Definition:Radius of Convergence

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 $f\left(z\right)$.

From the root test, it follows that if $\vert z-\xi\vertR$, then the series defining $f\left(z\right)$ is divergent.