Power Series and Radius of Convergence

Power series extend the concept of polynomials to infinite sums, providing a way to represent functions as limits of polynomial approximations with remarkable precision.

Definition of Power Series

Definition

A power series centered at $a$ is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$ where $c_0, c_1, c_2, \ldots$ are real (or complex) constants called the coefficients and $a$ is the center. The series defines a function of $x$ on whatever set it converges.

Definition

The radius of convergence $R$ of a power series $\sum c_n (x - a)^n$ is the value $R = \frac{1}{\limsup_{n \to \infty} |c_n|^{1/n}}$ with the convention $R = 0$ if the limsup is $\infty$ and $R = \infty$ if the limsup is $0$ . The series converges absolutely for $|x - a| < R$ and diverges for $|x - a| > R$ . The interval of convergence is the set of all $x$ where the series converges.

Determining the Radius

Theorem7.1Ratio Test for Radius

If the limit exists, the radius of convergence can be computed as $R = \lim_{n \to \infty} \left|\frac{c_n}{c_{n+1}}\right|$

ExampleCommon power series

$\sum_{n=0}^\infty x^n = \frac{1}{1-x}$ , radius $R = 1$
$\sum_{n=0}^\infty \frac{x^n}{n!} = e^x$ , radius $R = \infty$
$\sum_{n=1}^\infty \frac{x^n}{n} = -\ln(1 - x)$ , radius $R = 1$
$\sum_{n=0}^\infty n! \, x^n$ , radius $R = 0$ (converges only at $x = 0$ )

Behavior at Endpoints

RemarkEndpoint convergence requires individual analysis

The radius of convergence determines convergence in the open interval $(a - R, a + R)$ but says nothing about the endpoints $x = a \pm R$ . At each endpoint, convergence must be checked separately. For example, $\sum x^n/n$ converges at $x = -1$ (alternating series) but diverges at $x = 1$ (harmonic series), while $\sum x^n/n^2$ converges at both endpoints.

Power series form the foundation for Taylor and Maclaurin series, enabling the systematic representation of smooth functions as infinite polynomials.