How to calculate the series $\sum\limits_{n=1}^{\infty} \arctan(\frac{2}{n^{2}})$? I encountered the series 
$$
\sum_{n=1}^{\infty} \arctan\frac{2}{n^{2}}.
$$
I know it converges (by ratio test), but if I need to calculate its limit explicitly, how do I do that? Any hint would be helpful..

\begin{align*}
\sum_{n=1}^\infty\arctan\left ( \frac{2}{n^2} \right ) &=-arg \prod_{n=1}^\infty\left (1-\frac{2i}{n^2} \right ) \\ 
 &=-arg \prod_{n=1}^\infty\left (1-\frac{(\sqrt{2i})^2}{n^2} \right ) \\ 
 &=-arg\left(\frac{\sin(\pi\sqrt{2i})}{\pi\sqrt{2i}} \right ) \\ 
 &=-arg\left(-\frac{(1/2+i/2)\sinh\left(\pi \right )}{\pi} \right ) \\ 
 &= \frac{3\pi}{4}
\end{align*}
