Power Series

Summary

A power series $\sum a_n(x-c{)}^n$ converges inside an open interval (or disk) centered at $c$ of radius $R$ , and may or may not converge at the endpoints. The radius is often found with the ratio test.

Prerequisites

Ratio Test, Geometric Series, Absolute Convergence

Definition

Radius of Convergence

If the following limit exists,

with the conventions $R=0$ or $R=\mathrm{\infty }$ when the limit is $0$ or $\mathrm{\infty }$ . Equivalently, if $L=lim|a_{n+1}/a_n|$ , then $R=1/L$ when $L$ exists in $[0,\mathrm{\infty }]$ .

(Do not invert the ratio formula inconsistently: $R=lim|a_n/a_{n+1}|$ , not $1/R=lim|a_n/a_{n+1}|$ unless you define the limit the other way.)

Inside $|x-c|<R$ the series converges absolutely; outside $|x-c|>R$ it diverges. Endpoints $x=c\pm R$ require separate tests.

Worked Example

Consider $\sum _{n=0}^{\mathrm{\infty }}((x-2)/3{)}^n=\sum (1/3^n)(x-2{)}^n$ .

Here $a_n=3^{-n}$ , so

Absolute convergence for $|x-2|<3$ , i.e. $-1<x<5$ .

• At $x=-1$ : terms are $((-3)/3{)}^n=(-1{)}^n$ , which do not tend to $0$ → diverges.
• At $x=5$ : terms are $1^n=1\nrightarrow 0$ → diverges.

Interval of convergence: $(-1,5)$ .

Common Mistakes

• Including an endpoint without checking the resulting numerical series.
• Writing that $\sum (-1{)}^n$ converges (it does not).

Connections

• Taylor Series, Maclaurin Series, Binomial Series

References

Power series and radius of convergence are in OpenStax Calculus Volume 2.^[1]