Definition:Radius of Convergence

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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