I'd appreciate some help with this problem.
(a) Show that $\frac{1}{(1-e^z)^n} - \frac{1}{(1-e^z)^{n+1}} \quad (n=1,2,3,\ldots)$ has an anti-derivative whenever $e^z \neq 1$.
(b) Explain why $\oint_{C} \frac{1}{(1-e^z)^n}dz = \oint_{C} \frac{1}{(1-e^z)^{n+1}}dz$ for every natural number $n$ where $C$ is the circle $|z|=1$.
For (a) the term reduces to $\frac{-e^z}{(1-e^z)^{n+1}} = \frac{d}{dz}\left[\frac{1}{(1-e^z)^n}\right]$ for every natural number $n$ and is valid when $e^z \neq 1$. But how to prove it?
For (b) both integrands are discontinuous at $z=0$, so how do I proceed?