Radius of convergence

Definition

Over the complex numbers

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

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

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

It is defined as the largest $R$ such that the power series converges absolutely for all $z$ with $| z - z_{0} | < R$ and diverges for all $| z - z_{0} | > R$ .
It is given by the formula:

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

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 $\infty$ .

The open disk centered at $z_{0}$ and of radius equal to $R$ is termed the disk of convergence.

Over the real numbers

Consider the power series about a point $x_{0} \in R$ with coefficients $a_{n} \in R$ :

$\sum a_{n} (x - x_{0})^{n}$

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

It is defined as the largest $R$ such that the power series converges absolutely for all $x$ with $| x - x_{0} | < R$ and diverges for all $| x - x_{0} | > R$ .
It is given by the formula:

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

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 $\infty$ .

Facts

Power series is infinitely differentiable in disk of convergence: This result holds both over the reals and the complex numbers.