Existence of Radius of Convergence of Complex Power Series

Theorem
Let $\xi \in \C$.

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

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

Also see

 * Existence of Interval of Convergence of Power Series for a proof of the same result in real numbers.