Find the value of infinite series $\sum_{n=1}^{\infty} \tan^{-1}(2/n^2)$
Find the value of infinite series
$$ \sum_{n=1}^{\infty} \tan^{-1}\left(\frac{2}{n^2} \right) $$
I tried to find sequence of partial sums and tried to find the limit of that sequence. But I didn't get the answer.
It's the argument of the complex number $$\prod_{n=1}^\infty\left(1+\frac{2i}{n^2}\right) =\prod_{n=1}^\infty\frac{(n-1+i)(n+1-i)}{n^2} =\prod_{n=1}^\infty\frac{(n-1+i)(n^2+2n)}{n^2(n+1+i)}.$$ This infinite product telescopes.