How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$
Find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $