Solution Verification: Finding the Limit of a Sequence
The sequence ( x n ) (x_n) ( x n ) x 1 = 2 x_1 = 2 x 1 = 2 x n + 1 = x n + 1 n x_{n+1} = \sqrt{x_n + \frac{1}{n}} x n + 1 = x n + n 1 n ≥ 1 n \ge 1 n ≥ 1 x n n x_n^n x n n n n n
To begin, let's denote the limit of the sequence x n x_n x n n n n L L L ϵ \epsilon ϵ N N N n ≥ N n \ge N n ≥ N ∣ x n − L ∣ < ϵ |x_n - L| < \epsilon ∣ x n − L ∣ < ϵ
We can start by examining the recursive definition of the sequence. We have:
x n + 1 = x n + 1 n x_{n+1} = \sqrt{x_n + \frac{1}{n}}
x n + 1 = x n + n 1
Taking the n n n
x n + 1 n = ( x n + 1 n ) n 2 x_{n+1}^n = (x_n + \frac{1}{n})^{\frac{n}{2}}
x n + 1 n = ( x n + n 1 ) 2 n
Using the binomial theorem, we can expand the right-hand side as:
x n + 1 n = x n n 2 + n 2 x n n 2 − 1 ⋅ 1 n + ( n 2 ) ( n 2 − 1 ) 2 ! x n n 2 − 2 ⋅ ( 1 n ) 2 + … x_{n+1}^n = x_n^{\frac{n}{2}} + \frac{n}{2}x_n^{\frac{n}{2}-1} \cdot \frac{1}{n} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}x_n^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
x n + 1 n = x n 2 n + 2 n x n 2 n − 1 ⋅ n 1 + 2 ! ( 2 n ) ( 2 n − 1 ) x n 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of both sides as n n n
lim n → ∞ x n + 1 n = lim n → ∞ ( x n n 2 + n 2 x n n 2 − 1 ⋅ 1 n + ( n 2 ) ( n 2 − 1 ) 2 ! x n n 2 − 2 ⋅ ( 1 n ) 2 + … ) \lim_{n \to \infty} x_{n+1}^n = \lim_{n \to \infty} \left(x_n^{\frac{n}{2}} + \frac{n}{2}x_n^{\frac{n}{2}-1} \cdot \frac{1}{n} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}x_n^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots\right)
n → ∞ lim x n + 1 n = n → ∞ lim ( x n 2 n + 2 n x n 2 n − 1 ⋅ n 1 + 2 ! ( 2 n ) ( 2 n − 1 ) x n 2 n − 2 ⋅ ( n 1 ) 2 + … )
Using the fact that x n x_n x n L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Now, we can take the limit of the entire expression as n n n
lim n → ∞ x n + 1 n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … \lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
n → ∞ lim x n + 1 n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + …
Using the fact that L L L
\lim_{n \to \infty} x_{n+1}^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right<br/>
**Solution Verification: Finding the Limit of a Sequence - Q&A**
The sequence ( x n ) (x_n) ( x n ) x 1 = 2 x_1 = 2 x 1 = 2 x n + 1 = x n + 1 n x_{n+1} = \sqrt{x_n + \frac{1}{n}} x n + 1 = x n + n 1 n ≥ 1 n \ge 1 n ≥ 1 x n n x_n^n x n n n n n
Q: What is the recursive definition of the sequence ( x n ) (x_n) ( x n )
A: The sequence ( x n ) (x_n) ( x n ) x 1 = 2 x_1 = 2 x 1 = 2 x n + 1 = x n + 1 n x_{n+1} = \sqrt{x_n + \frac{1}{n}} x n + 1 = x n + n 1 n ≥ 1 n \ge 1 n ≥ 1
Q: What is the limit of the sequence x n n x_n^n x n n n n n
A: To find the limit of the sequence x n n x_n^n x n n n n n
Q: How do we simplify the expression for the limit of the sequence x n n x_n^n x n n
A: We can simplify the expression for the limit of the sequence x n n x_n^n x n n x n x_n x n L L L
Q: What is the final expression for the limit of the sequence x n n x_n^n x n n
A: The final expression for the limit of the sequence x n n x_n^n x n n
lim n → ∞ x n n = L n 2 + 1 2 L n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! L n 2 − 2 ⋅ ( 1 n ) 2 + … < / s p a n > < / p > < p > < s t r o n g > Q : H o w d o w e f i n d t h e v a l u e o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / s t r o n g > A : T o f i n d t h e v a l u e o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > , w e c a n u s e t h e f a c t t h a t < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > x < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > c o n v e r g e s t o < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > a n d t h e r e c u r s i v e d e f i n i t i o n o f t h e s e q u e n c e . < / p > < p > < s t r o n g > Q : W h a t i s t h e v a l u e o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / s t r o n g > A : T h e v a l u e o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < m o > = < / m o > < m s q r t > < m n > 2 < / m n > < / m s q r t > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L = 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.04 e m ; v e r t i c a l − a l i g n : − 0.0839 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.9161 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > < s t r o n g > Q : W h a t i s t h e f i n a l a n s w e r f o r t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / s t r o n g > A : T h e f i n a l a n s w e r f o r t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m u n d e r > < m r o w > < m i > l i m < / m i > < m o > < / m o > < / m r o w > < m r o w > < m i > n < / m i > < m o > → < / m o > < m i m a t h v a r i a n t = " n o r m a l " > ∞ < / m i > < / m r o w > < / m u n d e r > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < m o > = < / m o > < m s u p > < m s q r t > < m n > 2 < / m n > < / m s q r t > < m f r a c > < m i > n < / m i > < m n > 2 < / m n > < / m f r a c > < / m s u p > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < m s u p > < m s q r t > < m n > 2 < / m n > < / m s q r t > < m r o w > < m f r a c > < m i > n < / m i > < m n > 2 < / m n > < / m f r a c > < m o > − < / m o > < m n > 1 < / m n > < / m r o w > < / m s u p > < m o > + < / m o > < m f r a c > < m r o w > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m i > n < / m i > < m n > 2 < / m n > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m i > n < / m i > < m n > 2 < / m n > < / m f r a c > < m o > − < / m o > < m n > 1 < / m n > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < / m r o w > < m r o w > < m n > 2 < / m n > < m o s t r e t c h y = " f a l s e " > ! < / m o > < / m r o w > < / m f r a c > < m s u p > < m s q r t > < m n > 2 < / m n > < / m s q r t > < m r o w > < m f r a c > < m i > n < / m i > < m n > 2 < / m n > < / m f r a c > < m o > − < / m o > < m n > 2 < / m n > < / m r o w > < / m s u p > < m o > ⋅ < / m o > < m s u p > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m n > 1 < / m n > < m i > n < / m i > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m n > 2 < / m n > < / m s u p > < m o > + < / m o > < m o > … < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > lim n → ∞ x n n = 2 n 2 + 1 2 2 n 2 − 1 + ( n 2 ) ( n 2 − 1 ) 2 ! 2 n 2 − 2 ⋅ ( 1 n ) 2 + … < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.4144 e m ; v e r t i c a l − a l i g n : − 0.7 e m ; " > < / s p a n > < s p a n c l a s s = " m o p o p − l i m i t s " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6944 e m ; " > < s p a n s t y l e = " t o p : − 2.4 e m ; m a r g i n − l e f t : 0 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < s p a n c l a s s = " m r e l m t i g h t " > → < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > ∞ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n > < s p a n c l a s s = " m o p " > l i m < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.277 e m ; v e r t i c a l − a l i g n : − 0.0839 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.9161 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " > < s p a n s t y l e = " t o p : − 3.709 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.9161 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " > < s p a n s t y l e = " t o p : − 3.709 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.276 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.59 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m c l o s e " > ! < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.74 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6954 e m ; " > < s p a n s t y l e = " t o p : − 2.655 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6954 e m ; " > < s p a n s t y l e = " t o p : − 2.655 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > − < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.9161 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " > < s p a n s t y l e = " t o p : − 3.709 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > ⋅ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.604 e m ; v e r t i c a l − a l i g n : − 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.654 e m ; " > < s p a n s t y l e = " t o p : − 3.9029 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.123 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > … < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > < s t r o n g > Q : W h a t i s t h e s i m p l i f i e d e x p r e s s i o n f o r t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > ? < / s t r o n g > A : T h e s i m p l i f i e d e x p r e s s i o n f o r t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m u n d e r > < m r o w > < m i > l i m < / m i > < m o > < / m o > < / m r o w > < m r o w > < m i > n < / m i > < m o > → < / m o > < m i m a t h v a r i a n t = " n o r m a l " > ∞ < / m i > < / m r o w > < / m u n d e r > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < m o > = < / m o > < m s q r t > < m n > 2 < / m n > < / m s q r t > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > lim n → ∞ x n n = 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.4144 e m ; v e r t i c a l − a l i g n : − 0.7 e m ; " > < / s p a n > < s p a n c l a s s = " m o p o p − l i m i t s " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6944 e m ; " > < s p a n s t y l e = " t o p : − 2.4 e m ; m a r g i n − l e f t : 0 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < s p a n c l a s s = " m r e l m t i g h t " > → < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > ∞ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n > < s p a n c l a s s = " m o p " > l i m < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.04 e m ; v e r t i c a l − a l i g n : − 0.0839 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.9161 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > I n t h i s a r t i c l e , w e h a v e v e r i f i e d t h e s o l u t i o n t o t h e p r o b l e m o f f i n d i n g t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > a s < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > n < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > a p p r o a c h e s i n f i n i t y . W e h a v e u s e d t h e r e c u r s i v e d e f i n i t i o n o f t h e s e q u e n c e a n d t h e c o n c e p t o f l i m i t s t o s i m p l i f y t h e e x p r e s s i o n f o r t h e l i m i t o f t h e s e q u e n c e . W e h a v e a l s o f o u n d t h e v a l u e o f < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > L < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > L < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > L < / s p a n > < / s p a n > < / s p a n > < / s p a n > a n d t h e f i n a l a n s w e r f o r t h e l i m i t o f t h e s e q u e n c e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m i > x < / m i > < m i > n < / m i > < m i > n < / m i > < / m s u b s u p > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > x n n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v e r t i c a l − a l i g n : − 0.247 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > . < / p > \lim_{n \to \infty} x_n^n = L^{\frac{n}{2}} + \frac{1}{2}L^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}L^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
</span></p>
<p><strong>Q: How do we find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>?</strong>
A: To find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>, we can use the fact that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> converges to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> and the recursive definition of the sequence.</p>
<p><strong>Q: What is the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>?</strong>
A: The value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">L = \sqrt{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0839em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9561em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.9161em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.0839em;"><span></span></span></span></span></span></span></span></span></span></p>
<p><strong>Q: What is the final answer for the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>?</strong>
A: The final answer for the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo></mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup><mo>=</mo><msup><msqrt><mn>2</mn></msqrt><mfrac><mi>n</mi><mn>2</mn></mfrac></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><msqrt><mn>2</mn></msqrt><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mfrac><mrow><mrow><mo fence="true">(</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>−</mo><mn>1</mn><mo fence="true">)</mo></mrow></mrow><mrow><mn>2</mn><mo stretchy="false">!</mo></mrow></mfrac><msup><msqrt><mn>2</mn></msqrt><mrow><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>−</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mo fence="true">(</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow><annotation encoding="application/x-tex">\lim_{n \to \infty} x_n^n = \sqrt{2}^{\frac{n}{2}} + \frac{1}{2}\sqrt{2}^{\frac{n}{2}-1} + \frac{\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)}{2!}\sqrt{2}^{\frac{n}{2}-2} \cdot \left(\frac{1}{n}\right)^2 + \ldots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4144em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.277em;vertical-align:-0.0839em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9561em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.9161em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.276em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9561em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.9161em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<p><strong>Q: What is the simplified expression for the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>?</strong>
A: The simplified expression for the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo></mo></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\lim_{n \to \infty} x_n^n = \sqrt{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4144em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0839em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9561em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.9161em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
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c5.3,-9.3,12,-14,20,-14
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s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.0839em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>In this article, we have verified the solution to the problem of finding the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> approaches infinity. We have used the recursive definition of the sequence and the concept of limits to simplify the expression for the limit of the sequence. We have also found the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> and the final answer for the limit of the sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">x_n^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9114em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>.</p>
n → ∞ lim x n n = L 2 n + 2 1 L 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) L 2 n − 2 ⋅ ( n 1 ) 2 + … < / s p an >< / p >< p >< s t ro n g > Q : Ho w d o w e f in d t h e v a l u eo f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > ? < / s t ro n g > A : T o f in d t h e v a l u eo f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > , w ec an u se t h e f a c tt ha t < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > x < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > co n v er g es t o < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > an d t h erec u rs i v e d e f ini t i o n o f t h ese q u e n ce . < / p >< p >< s t ro n g > Q : Wha t i s t h e v a l u eo f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > ? < / s t ro n g > A : T h e v a l u eo f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > i s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > L < / mi >< m o >=< / m o >< m s q r t >< mn > 2 < / mn >< / m s q r t >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L = 2 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.04 e m ; v er t i c a l − a l i g n : − 0.0839 e m ; " >< / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.9161 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p >< s t ro n g > Q : Wha t i s t h e f ina l an s w er f or t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > ? < / s t ro n g > A : T h e f ina l an s w er f or t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m u n d er >< m ro w >< mi > l im < / mi >< m o > < / m o >< / m ro w >< m ro w >< mi > n < / mi >< m o >→< / m o >< mima t h v a r ian t = " n or ma l " > ∞ < / mi >< / m ro w >< / m u n d er >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< m o >=< / m o >< m s u p >< m s q r t >< mn > 2 < / mn >< / m s q r t >< m f r a c >< mi > n < / mi >< mn > 2 < / mn >< / m f r a c >< / m s u p >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< m s u p >< m s q r t >< mn > 2 < / mn >< / m s q r t >< m ro w >< m f r a c >< mi > n < / mi >< mn > 2 < / mn >< / m f r a c >< m o > − < / m o >< mn > 1 < / mn >< / m ro w >< / m s u p >< m o > + < / m o >< m f r a c >< m ro w >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< mi > n < / mi >< mn > 2 < / mn >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< mi > n < / mi >< mn > 2 < / mn >< / m f r a c >< m o > − < / m o >< mn > 1 < / mn >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< / m ro w >< m ro w >< mn > 2 < / mn >< m os t re t c h y = " f a l se " > ! < / m o >< / m ro w >< / m f r a c >< m s u p >< m s q r t >< mn > 2 < / mn >< / m s q r t >< m ro w >< m f r a c >< mi > n < / mi >< mn > 2 < / mn >< / m f r a c >< m o > − < / m o >< mn > 2 < / mn >< / m ro w >< / m s u p >< m o > ⋅ < / m o >< m s u p >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< mn > 1 < / mn >< mi > n < / mi >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< mn > 2 < / mn >< / m s u p >< m o > + < / m o >< m o > … < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > n → ∞ lim x n n = 2 2 n + 2 1 2 2 n − 1 + 2 ! ( 2 n ) ( 2 n − 1 ) 2 2 n − 2 ⋅ ( n 1 ) 2 + … < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.4144 e m ; v er t i c a l − a l i g n : − 0.7 e m ; " >< / s p an >< s p an c l a ss = " m o p o p − l imi t s " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6944 e m ; " >< s p an s t y l e = " t o p : − 2.4 e m ; ma r g in − l e f t : 0 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< s p an c l a ss = " m re l m t i g h t " >→< / s p an >< s p an c l a ss = " m or d m t i g h t " > ∞ < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an >< s p an c l a ss = " m o p " > l im < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.277 e m ; v er t i c a l − a l i g n : − 0.0839 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.9161 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " >< s p an s t y l e = " t o p : − 3.709 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.9161 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " >< s p an s t y l e = " t o p : − 3.709 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< s p an c l a ss = " mbinm t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.276 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.59 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m c l ose " > ! < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.74 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6954 e m ; " >< s p an s t y l e = " t o p : − 2.655 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6954 e m ; " >< s p an s t y l e = " t o p : − 2.655 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > − < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.9161 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.1931 e m ; " >< s p an s t y l e = " t o p : − 3.709 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6915 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< s p an c l a ss = " mbinm t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > ⋅ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.604 e m ; v er t i c a l − a l i g n : − 0.95 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.654 e m ; " >< s p an s t y l e = " t o p : − 3.9029 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.123 e m ; " >< / s p an >< s p an c l a ss = " minn er " > … < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p >< s t ro n g > Q : Wha t i s t h es im pl i f i e d e x p ress i o n f or t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > ? < / s t ro n g > A : T h es im pl i f i e d e x p ress i o n f or t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m u n d er >< m ro w >< mi > l im < / mi >< m o > < / m o >< / m ro w >< m ro w >< mi > n < / mi >< m o >→< / m o >< mima t h v a r ian t = " n or ma l " > ∞ < / mi >< / m ro w >< / m u n d er >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< m o >=< / m o >< m s q r t >< mn > 2 < / mn >< / m s q r t >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > n → ∞ lim x n n = 2 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.4144 e m ; v er t i c a l − a l i g n : − 0.7 e m ; " >< / s p an >< s p an c l a ss = " m o p o p − l imi t s " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6944 e m ; " >< s p an s t y l e = " t o p : − 2.4 e m ; ma r g in − l e f t : 0 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< s p an c l a ss = " m re l m t i g h t " >→< / s p an >< s p an c l a ss = " m or d m t i g h t " > ∞ < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an >< s p an c l a ss = " m o p " > l im < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.04 e m ; v er t i c a l − a l i g n : − 0.0839 e m ; " >< / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9561 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.9161 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.0839 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > I n t hi s a r t i c l e , w e ha v e v er i f i e d t h eso l u t i o n t o t h e p ro b l e m o ff in d in g t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > a s < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > n < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< / s p an > a pp ro a c h es in f ini t y . W e ha v e u se d t h erec u rs i v e d e f ini t i o n o f t h ese q u e n ce an d t h eco n ce pt o f l imi t s t os im pl i f y t h ee x p ress i o n f or t h e l imi t o f t h ese q u e n ce . W e ha v e a l so f o u n d t h e v a l u eo f < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > L < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > L < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > L < / s p an >< / s p an >< / s p an >< / s p an > an d t h e f ina l an s w er f or t h e l imi t o f t h ese q u e n ce < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b s u p >< mi > x < / mi >< mi > n < / mi >< mi > n < / mi >< / m s u b s u p >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > x n n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.9114 e m ; v er t i c a l − a l i g n : − 0.247 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6644 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > . < / p >