TITLE: Prove that $\zeta(4)=\pi^4/90$
QUESTION [9 upvotes]: I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$
I think I can do this given the Laurent series for $\cot z$ centered at the origin, but I don't know how to find the first few terms of the Laurent series (I can use Cauchy's Integral Formula to find the first coefficient).

REPLY [4 votes]: The calculus of residues solution comes from integrating $\frac{\pi\cot{\pi z}}{z^4}$ counterclockwise around the appropriate curve. The solution is worked out here.