Find $\frac{\sum\limits^{2499}_{k=1}{\sqrt{10+\sqrt{50+\sqrt{k}}}}}{\sum\limits^{2499}_{k=1}{\sqrt{10-\sqrt{50+\sqrt{k}}}}}$
135968
Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$
116649
Value of: $ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$
99420
Compute $\sum_{n=1}^\infty \frac{1}{(n+1)\sqrt[]{n}+n\sqrt[]{n+1}}$
89367
Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $
137027
Find the value of $\sum_{n=0}^\infty\frac{1}{9n^2+9n+2}$
14883
Find $\sum_{n=1}^{\infty}$ $\frac{n^{2}}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)}$
93984
How find this Fibonacci sequence sum $\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}$
47706
Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $
143753
Please help me to find the sum $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$
58262
Find the sum of the series $1+\frac{1}{3}\cdot\frac{1}{4}+\frac{1}{5}\cdot\frac{1}{4^2}+\frac{1}{7}\cdot\frac{1}{4^3}+\cdots$
74120
Find the value of $\sum_{n=1}^{\infty} \frac{2}{n}-\frac{4}{2n+1}$
81589
Find the sum of $\sum_{k=0}^{\infty} 1/[(r+2k+2)(s+2k+2)]$
60061
Find the value of : $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$
89263
Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$
136066
Find the sum $\sum _{n=1}^{\infty }\left(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}\right)$
31174
Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
84159
The sum to n terms of the series $\frac{1}{\sqrt 1+\sqrt 3} + \frac{1}{\sqrt 3 +\sqrt 5} ....$