Here is a problem I came across while studying for an upcoming exam:
Suppose that $f$ is analytic in the region $|z| < R$, except for a simple pole at $z_0$ with $0 < |z_0| < R$. Let $f(z) = \sum_{n=0}^\infty a_n z^n$ be the Taylor series of $f$ at the origin. Show that the following limit exists and is not $0$: $A = \lim_{n\to \infty}a_n z_0^{n+1}$.
I am not sure how to approach this at all. I would love a hint to get me started...thanks in advance