Limit of Taylor Coefficients Near a Simple Pole

Question

I am practicing for an analysis qual and came across this question:

Suppose $f$ is analytic on $|z| < R$ except for a simple pole at $z_0$, $0 < \> |z_0| < R$. If $f(z) = \sum_{n=0}^\infty a_n z^n$ is the Taylor series for $f$ near the origin, show that $$ A = \lim_{n \to \infty} a_n \> z_0^{n+1} $$ exists and is not zero.

I am stumped on how to start this and would appreciate any hints.

Answer 1

(I just used this fact today in a research meeting!)

Hint: if $r$ is the residue of $f(z)$ at $z=z_0$, consider the Maclaurin series coefficients of $g(z) = f(z) - r/(z-z_0)$.