Radius of convergence

Definition

Consider the power series about a point $z_{0}\in \mathbb {C}$ with coefficients $a_{n}\in \mathbb {C}$:

$\sum a_{n}(z-z_{0})^{n}$

The radius of convergence of this power series is defined in the following equivalent ways:

• It is given by the formula:

$R={\frac {1}{\lim \sup \left|a_{n}\right|^{1/n}}}$

If the denominator is $\infty$, the radius of convergence is defined as 0, and if the denominator is 0, the radius of convergence is taken to be $\infty$.