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Section 6.1 Power Series and Radius of Convergence

A power series is a series whose terms are powers of \(x-x_0\text{.}\) The root test from Proposition 3.2.12 shows that such series have much more rigid convergence behavior than an arbitrary series: there is always a single radius inside which the series converges absolutely and outside which it diverges.

Subsection 6.1.1

A power series centered at \(x_0 \in \R\) is a series of the form
\begin{equation*} \sum_{n=0}^\infty a_n(x-x_0)^n. \end{equation*}
It always converges at the center \(x=x_0\text{.}\) If it converges at some point \(x \ne x_0\text{,}\) we call it a convergent power series; otherwise it is a divergent power series.

Example 6.1.1.

Every radius of convergence can occur.
  1. The series
    \begin{equation*} \sum_{n=0}^\infty n!x^n \end{equation*}
    has radius of convergence \(0\text{.}\) Indeed, \((n!)^{1/n}\to \infty\) because \(n! \ge (n/2)^{n/2}\) for large \(n\text{.}\)
  2. The geometric series
    \begin{equation*} \sum_{n=0}^\infty x^n \end{equation*}
    has radius of convergence \(1\) by Proposition 3.2.1.
  3. The exponential series
    \begin{equation*} \sum_{n=0}^\infty \frac{x^n}{n!} \end{equation*}
    has radius of convergence \(\infty\text{,}\) because the ratio test gives
    \begin{equation*} \frac{|x|^{n+1}/(n+1)!}{|x|^n/n!}=\frac{|x|}{n+1}\to 0 \end{equation*}
    for every \(x \in \R\text{.}\)

Proof.
Fix \(x \in \R\) and apply the root test to the series
\begin{equation*} \sum_{n=0}^\infty a_n(x-x_0)^n. \end{equation*}
Since
\begin{equation*} \limsup |a_n(x-x_0)^n|^{1/n} = |x-x_0| \limsup |a_n|^{1/n}, \end{equation*}
the root test shows that the series converges absolutely when this quantity is less than \(1\) and diverges when it is greater than \(1\text{.}\) Defining \(R\) by the displayed formula gives the result.
The number \(R\) is called the radius of convergence of the power series. The interval
\begin{equation*} (x_0-R,x_0+R) \end{equation*}
is called its interval of convergence. The theorem says nothing about the boundary points \(x_0-R\) and \(x_0+R\text{;}\) these must be checked separately.

Example 6.1.3.

Boundary behavior can vary even when the radius of convergence is the same. Each of the following power series has radius \(1\text{.}\)
  1. \(\sum_{n=0}^\infty x^n\) diverges at both endpoints \(x=\pm 1\) by Proposition 3.2.1.
  2. \(\sum_{n=1}^\infty x^n/n\) diverges at \(x=1\) by the \(p\)-series test (Proposition 3.2.5) but converges at \(x=-1\) by the alternating series test (Proposition 3.2.15).
  3. \(\sum_{n=1}^\infty x^n/n^2\) converges absolutely at both endpoints by Proposition 3.2.5.
By translating the variable, one may often reduce questions about a power series centered at \(x_0\) to a power series centered at \(0\text{.}\) For that reason, many examples below are written in the simpler form \(\sum a_nx^n\text{.}\)
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