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:

$R=\frac{1}{limsup{\left|a_n\right|}^{1/n}}$

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