Derivative of Complex Power Series

Theorem
Let $\xi \in \C$ be a complex number.

Let $\sequence {a_n}$ be a sequence in $\C$.

Let $\ds \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a power series in a complex variable $z \in \C$ about $\xi$.

Let $R$ be the radius of convergence of the series defining $\map f z$.

Let $\cmod {z - \xi} < R$.

Then $f$ is complex-differentiable and its derivative is:
 * $\ds \map {f'} z = \sum_{n \mathop = 1}^\infty n a_n \paren {z - \xi}^{n - 1}$