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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?

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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 $z=z_0$.

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