Definition:Radius of Convergence/Complex Domain

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

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

be a power series about $\xi$.

The radius of convergence is the extended real number $R \in \overline{\R}$ defined by:


 * $R = \displaystyle \inf \, \left\{{ \left\vert{z - \xi}\right\vert : z \in \C, \sum_{n \mathop = 0}^\infty a_n \left({z - \xi}\right)^n \text{ is divergent}}\right\}$

where a divergent series is a series that is not convergent.

As usual, $\inf \varnothing = +\infty$.

Also see

 * Radius of Convergence is Radius, which shows 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.