The Bernoulli numbers \(B_k\) are defined in terms of the Laurent series \[ \frac {1}{e^z-1} =\frac 1z-\frac 12+\sum _{k=1}^{\infty }(-1)^{k-1}\frac {B_k}{(2k)!}z^{2k-1}. \]
(a) Find the Laurent series of \(\cot \pi z\) in terms of the Bernoulli numbers.
(b) Using (a) and the fact on partial fractions \[ \pi \cot \pi z=\frac 1z+\sum _{n=1}^{\infty }\frac {2z}{z^2-n^2} \] to prove \[ \sum _{n=1}^{\infty }\frac {1}{n^{2k}} =2^{2k-1}\frac {B_k}{(2k)!}\pi ^{2k} \] for any positive integer \(k\).