# Describe the $a_n$ in terms of the derivatives of $f$ given by a convergent power series at $z_0$.

Suppose $$f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n$$ is given by a convergent power series around $$z_0$$. Describe the $$a_n$$ in terms of the derivatives of $$f$$ at $$z_0$$.

I know that the derivative of $$f(z)$$ is $$\sum_{n=0}^\infty na_n(z-z_0)^{n-1}$$ on the disc $$\{z: |z-z_0|\lt R\}$$ ($$R$$ is the radius of convergence), but I had trouble with how to represent $$a_n$$ by above derivative. Any hints?

This is standard text book material. The formula is $$a_n=\ \frac {f^{(n)}(z_0)} {n!}$$ and it is obtained by repeatedly differentiating the series and putting $$z=z_0$$.