Definition:Radius of Convergence/Complex Domain

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

For $z \in \C$, let:
 * $\displaystyle f \left({z}\right) = \sum_{n=0}^\infty a_n \left({z - \xi}\right)^n$

be a power series about $\xi$.

Let:
 * $\displaystyle \rho = \limsup_{n\to\infty} \vert a_n \vert^{1/n}$

Then $R = \rho^{-1}$ is called the radius of convergence of the series defining $f \left({z}\right)$.

Also see
From the root test, it follows that:
 * if $\left \vert {z - \xi}\right \vert < R$, then the power series defining $f \left({z}\right)$ is absolutely convergent
 * if $\left \vert {z - \xi}\right \vert > R$, then the power series defining $f \left({z}\right)$ is divergent.