Derivative of Complex Power Series

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

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

Let $\displaystyle f \left({z}\right) = \sum_{n \mathop =0}^\infty a_n \left({z - \xi}\right)^n$ be a power series in a complex variable $z \in \C$ about $\xi$.

Let $R$ be the radius of convergence of the series defining $f \left({z}\right)$.

Let $\left \vert{z - \xi}\right \vert < R$.

Then $f$ is complex differentiable and its derivative is:
 * $\displaystyle f’ \left({z}\right) = \sum_{n \mathop =1}^\infty n a_n \left({z - \xi}\right)^{n-1}$