How to prove the following using residue theorem?

Question

I have this HW and I don't know how to approach it, does anybody know how can it be shown that $$∑_{n=1}^∞\frac{1}{n^2} =\frac{π^2}{6}$$ using residue theorem?

Answer 1

In four dense pages in this paper the author starts with the residue theorem on page 7 and proves that $\zeta(2)=\pi^2/6.$ I skimmed the article and it looks like a clear exposition to me. There are certainly other versions.

The paper is by Brendan Sullivan, Numerous Proofs of $\zeta(2)=\frac{\pi^2}{6},$ dated 2013.