Existence of Radius of Convergence of Complex Power Series

Theorem
Let $\xi \in \C$.

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

Then there exists a radius of convergence $R \in \overline \R$ of $S \paren z$.

Also see

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