How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$
7177
Find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $
38760
Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
45440
$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$ c0nvergence
126264
Does the series $\sum \limits _{n=0}^{\infty} \cos(n\pi)$ converge or diverge?
32866
How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$
125933
On the series $\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{n \log n}$
54661
How to find the sum $\sum\limits_{n=1}^{\infty}\frac{1}{n!(n^4+n^2+1)}$
137155
Sum $ \sum\limits_{k=1}^{\infty} \frac{k^2}{2^k}$
27224
Infinite series $\sum_{k=1}^\infty \frac{k}{2^k}$ and $\sum_{k=1}^\infty \frac{k^2}{2^k}$
119147
How to prove that $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?
135380
Show that $\sum_{n=1}^\infty \frac{n^2}{(n+1)!}=e-1$
114673
Proving the convergence of $\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$
67752
Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$
5650
Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$
25504
How to find sum of the infinite series $\sum_{n=1}^{\infty} \frac{1}{ n(2n+1)}$
59485
Infinite Series: $\sum_{n=1}^\infty \frac{1}{2^nn}$
102531
Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-