I am very new to this kind of thing, so I would really appreciate at least a little hint:

Main goal is to evaluate

$\displaystyle \displaystyle \[\sum\limits_{n = 1}^{ + \infty } {\arctan \left( {\frac{1}{{2 \cdot {n^2}}}} \right)} \]$

and in the process to come up with the partial summation formula.

I actually figured out the nature of adding two arctan() results

$\displaystyle \displaystyle \[\arctan \left( x \right) + \arctan \left( y \right) = \arctan \left( {\frac{{x + y}}{{1 - x \cdot y}}} \right)\]$

but still not able to move forward...