# Definition:Radius of Convergence

*This page is about the radius of convergence of a power series. For other uses, see Definition:Radius.*

## Definition

### Real Domain

Let $\xi \in \R$ be a real number.

Let $\displaystyle S \paren x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.

Let $I$ be the interval of convergence of $S \paren x$.

Let the endpoints of $I$ be $\xi - R$ and $\xi + R$.

(This follows from the fact that $\xi$ is the midpoint of $I$.)

Then $R$ is called the **radius of convergence** of $S \paren x$.

If $S \paren x$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.

### Complex Domain

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

For $z \in \C$, let:

- $\displaystyle f \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$

be a power series about $\xi$.

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

- $R = \displaystyle \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$

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

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

## Linguistic Note

The plural of **radius** is **radii**, pronounced ** ray-dee-eye**.

This irregular plural form stems from the Latin origin of the word **radius**, meaning **ray**.

The ugly incorrect form **radiuses** can apparently be found, but rarely in a mathematical context.

## Sources

- 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.3.2$: Power series: Theorem $1.11$