$\zeta(2n)$ proof

Question

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_{2k}(2\pi)^{2k}}{2(2k)!} \end{equation}

I know how we start by looking at the product of sine and use the generatinf function for the Bernoulli numbers to connect them. I am finding it hard to find a source that doesn't just assume the result or say that it is fairly trivial.

Any help would be appreciated, Thanks

Answer 1

There is a nice proof in Remmert's book "Funktionentheorie" using residue calculus. Also, Tom M. Apostol has given a short elementary proof in The American Mathematical Monthly 1973 (JSTOR, Google), based on quadratic cotangent identities. The article also surveys other proofs of this famous identity, and gives many useful references.

Answer 2

This happens to have been the topic of an article in the May 2015 American Mathematical Monthly. You can get access here.

Edit: instead of me sumerizing the article, you can find it here for free (thanks to Raymond Manzoni for the link).