Power Series

Power series are series of the form $\sum a_n x^n$ that define functions within a radius of convergence. They converge absolutely and uniformly on compact subsets of their interval of convergence. Power series can be differentiated and integrated term-by-term, making them powerful tools for solving differential equations and approximating functions.

Definition and radius of convergence

Definition7.1Power series

A power series centered at $x_0$ is a series of the form

$\sum_{n=0}^\infty a_n (x - x_0)^n,$

where $(a_n)$ are coefficients. The radius of convergence $R$ is

$R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}.$

The series converges absolutely for $|x - x_0| < R$ and diverges for $|x - x_0| > R$ .

ExampleGeometric series

$\sum x^n$ has $a_n = 1$ , so $R = 1$ . The series converges for $|x| < 1$ to $1/(1-x)$ and diverges for $|x| > 1$ .

Examplee^x = Σ x^n/n!

For $\sum x^n/n!$ , we have $a_n = 1/n!$ , so $|a_n|^{1/n} = 1/(n!)^{1/n} \to 0$ . Thus $R = \infty$ , and the series converges for all $x \in \mathbb{R}$ .

Term-by-term operations

Theorem7.1Term-by-term differentiation

If $f(x) = \sum a_n x^n$ has radius of convergence $R > 0$ , then $f$ is differentiable on $(-R, R)$ , and

$f'(x) = \sum_{n=1}^\infty n a_n x^{n-1}.$

The derived series has the same radius of convergence $R$ .

Theorem7.2Term-by-term integration

$\int_0^x f(t) \, dt = \sum_{n=0}^\infty \frac{a_n}{n+1} x^{n+1}.$

Examplearctan(x) = Σ (-1)^n x^(2n+1)/(2n+1)

Starting from $1/(1+x^2) = \sum (-1)^n x^{2n}$ for $|x| < 1$ , integrate term-by-term:

$\arctan(x) = \int_0^x \frac{1}{1+t^2} \, dt = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}.$

Summary

Power series are flexible tools for defining and analyzing functions:

Radius of convergence determined by $\limsup |a_n|^{1/n}$ .
Converge uniformly on compact subsets of $|x - x_0| < R$ .
Term-by-term differentiation and integration allowed within the radius.
Applications: Taylor series, solving ODEs, approximating functions.

See Taylor's Theorem and Uniform Convergence.