Complex analysis to evaluate series
According to a theorem I have seen in class, we have
It can easily be seen that the theorem works for , but I've been trying to use it to find the exact value of without success. The poles of fare obviously i and -i, for which the residues are respectively and , which implies that
Of course, f is an even function, so .
But, when I approximate the sum in Maple or MATLAB, I get another answer (which is the right one): -0.3639849730...
Does anyone have an idea of what I have done wrong?
Thank you all!!