Prove $\sum\limits_{n\mathop=0}^{\infty}\frac{1}{n^2+1}=\frac{1}{2}+\frac{\pi}{2}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$
28378
Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$
145019
Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$
136066
Find the sum $\sum _{n=1}^{\infty }\left(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}\right)$
9610
Find the sum of $\sum\limits_{n=1}^{\infty} \frac{x^{n}}{n(n+1)}$
11622
Find $\lim _{n\to \infty }\left(\frac{1}{n^2+1}+\frac{1}{n^2+2}+...+\frac{1}{n^2+n}\right)$
158737
What is the value of the series $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n^2}$?
157539
Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$
62443
Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
14024
Convergent series and divergent series $\sum\limits_{n=1}^\infty \frac{n}{n^3+1}$ and $\sum\limits_{n=1}^\infty \frac{n^2+1}{n^3+1}$
78167
Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges
107789
Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$
60031
Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
71240
The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$
59253
Summing the power series $\sum\limits_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}\prod\limits_{k=1}^n\frac{2k-1}{2k} $
114031
Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
38760
Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
5650
Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$