Existence of Radius of Convergence of Complex Power Series

Theorem

Let $\xi \in \C$.

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

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

Absolute Convergence

Let $\map {B_R} \xi$ denote the open $R$-ball of $\xi$.

Let $z \in \map {B_R} \xi$.

Then $\map S z$ converges absolutely.

If $R = +\infty$, we define $\map {B_R} \xi = \C$.

Divergence

Let $\map { {B_R}^-} \xi$ denote the closed $R$-ball of $\xi$.

Let $z \notin \map { {B_R}^-} \xi$.

Then $\map S z$ is divergent.