Complex analysis to evaluate series

Hey all,

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 *f*are obviously *i* and *-i*, for which the residues are respectively and , which implies that

Of course, *f* is an even function, so .

Therefore,

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!!