I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory.
I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of $f(z)$
$$\sum_1^\infty \frac{1}{n^2}=\frac{1}{2}\sum_{-\infty}^\infty \frac{1}{n^2}$$. Now the only pole is $z=0$ which is a double pole, but when I evaluate:
$$\frac{d}{dz} \pi \cot (\pi z) \frac{1}{z^2}(z^2)=\infty$$
Can somebody show me why my method doesn't work?
