Discussion on question by onepound: How to compute the residue of $f(z)=\frac{n/z}{z^{n}-1}$ to be $-n$ at $z=0$?

Imported from a comment discussion on https://mathematica.stackexchange.com/questions/280534/how-to-compute-the-residue-of-fz-fracn-zzn-1-to-be-n-at-z-0