Definition:Radius of Convergence

Definition

Real Domain

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

Let $\displaystyle S \left({x}\right) = \sum_{n=0}^\infty a_n \left({x - \xi}\right)^n$ be a power series about $\xi$.

Let $I$ be the interval of convergence of $S \left({x}\right)$.

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 $S \left({x}\right)$.

If $S \left({x}\right)$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.

Definition

<onlyinclude> Let $\xi \in \C$ be a complex number.

For $z \in \C$, let:

$\displaystyle f \left({z}\right) = \sum_{n=0}^\infty a_n \left({z - \xi}\right)^n$

be a power series about $\xi$.

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

$R = \displaystyle \inf \, \left\{{ \left\vert{z - \xi}\right\vert : z \in \C, \sum_{n \mathop = 0}^\infty a_n \left({z - \xi}\right)^n \text{ is divergent}}\right\}$

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

Also see

From the root test, it follows that:

if $\left \vert {z - \xi}\right \vert < R$, then the power series defining $f \left({z}\right)$ is absolutely convergent

if $\left \vert {z - \xi}\right \vert > R$, then the power series defining $f \left({z}\right)$ is divergent.