Prove $\sum_{n=1}^{\infty} \frac{\frac{\pi}{2}-\text{Si}(n \pi)}{n}=\frac{\pi}{2}\left(1-\ln{\pi}\right)$
129734
Computing $\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$
55339
Find the sum $\sum_{n=1}^{\infty}\frac{ 1}{n^{2}(2n-1)}$
67752
Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$
74286
Prove: $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$
106281
Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$
63968
How to prove that $\lim\limits_{n\rightarrow \infty} \frac{1}{n^2}\sum\limits_{k=1}^{n}(n \bmod k)=1-\frac{\pi^2}{12}$?
114031
Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
24575
How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?
8913
Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
150563
Find the sum $\sum_{n=1}^{\infty} \frac{\ln n}{n(n- \ln n)}$
119147
How to prove that $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?
150902
Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$
62443
Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
132126
Proving $\sum\limits_{n=1}^{\infty } \frac{\mu (n)}n=0$ where $\mu$ is the Möbius function
30407
Prove that $\sum_{n=1}^\infty\frac{n}{n^2+1}$ is divergent
44970
Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$
150148
Sum of $\sum^\infty_{n=2} \frac{1}{\ln{(n!)}}$