How do I evaluate the Complex Integral $(z^{-n})/(e^z−1) $ using residue theory? Over here we had a similar looking question:
How do I evaluate the Complex Integral $(z^n)/(e^z - 1)$ using residue theory?
and the answer was that (rather unremarkably) we could remove the singularity made by $1/(e^z-1)$ for $z=0$ and conclude Res$((z^n)/(e^z-1))=0$ for all $n>0$ and Res$(1/(e^z-1))=1$.
I think this residue calculation becomes rather more interesting when we put a $-$ in front of the $n$. Then we have an (n+1)th order pole at $z=0$. Can we calculate the residue other than with this limit formula?

The coefficients of the power series $z/(e^z-1)$ are by definition the Bernoulli numbers (divided by a factorial), so they are the residues of your function.
Now the question remains what you mean by "calculate", i.e. what kind of answer do you seek?
