The Maclaurin Series for holomorphic function f

Question

$\textbf{Exercise} \quad $Let $R>1$ and let $f$ be holomorphic on $\vert z \vert <R$ except at $z=1$, where $f$ has a simple pole. If \begin{align*} f(z)=\sum_{n=0}^{\infty}a_nz^n \quad (\vert z \vert <1) \end{align*} is the Maclaurin series for $f$, show that $\lim_{n\rightarrow \infty} a_n$ exists.

We knew that $a_n= f^n(0)/n!$. But, I don't know how to use fact $f$ has a simple pole at $z=1$.

Any help is appreciated ... Thank you!

Answer 1

Hint: $(1-z)f(z)$ exends to be holomorphic in $D(0,R).$ This implies $\sum_{n=1}^{\infty}(a_n-a_{n-1})z^n$ has radius of convergence at least $R.$