Radius of convergence

Definition

Over the complex numbers

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 defined as the largest $R$ such that the power series converges absolutely for all $z$ with $\left|z-z_{0}\right|<R$ and diverges for all $\left|z-z_{0}\right|>R$.
• 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 $\infty$.

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 $x_{0}\in \mathbb {R}$ with coefficients $a_{n}\in \mathbb {R}$:

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

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 $\left|x-x_{0}\right|<R$ and diverges for all $\left|x-x_{0}\right|>R$.
• 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 $\infty$.

Facts

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