Skip to content
Mom's Basement University

Power Series

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what a power series is, how to find where it converges, and why power series are one of the most useful tools in all of mathematics.

The Concept

Section titled “The Concept”

A power series centered at a is an infinite polynomial

∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+c3(x−a)3+⋯\sum_{n=0}^{\infty} c_n(x - a)^n = c_0 + c_1(x - a) + c_2(x - a)^2 + c_3(x - a)^3 + \cdots

The coefficients c₀, c₁, c₂, … are constants, and a is the center. When a = 0, you get the simpler form

∑n=0∞cnxn=c0+c1x+c2x2+⋯\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots

Radius and Interval of Convergence

Section titled “Radius and Interval of Convergence”

Every power series has a radius of convergence R such that

converges when ∣x−a∣<Rdiverges when ∣x−a∣>R\text{converges when } |x - a| \lt R \qquad \text{diverges when } |x - a| \gt R

R can be 0 (converges only at the center), a finite positive number, or infinity (converges everywhere).

To find R, apply the ratio test to the absolute values of the terms. At the endpoints where the distance from center equals exactly R, you have to test each one separately using the convergence tests from previous lessons.

The interval of convergence is the full set of x values where the series converges, including any endpoints that pass their individual tests.

Worked Examples

Section titled “Worked Examples”

Example 1: Radius of convergence equals zero

Find the radius of convergence of

∑n=0∞n! xn\sum_{n=0}^{\infty} n!\,x^n

Apply the ratio test

L=lim⁡n→∞∣(n+1)! xn+1n! xn∣=lim⁡n→∞(n+1)∣x∣=∞L = \lim_{n \to \infty} \left|\frac{(n+1)!\,x^{n+1}}{n!\,x^n}\right| = \lim_{n \to \infty} (n+1)|x| = \infty

For any x other than 0, the limit is infinity, which is greater than 1. The series only converges at x = 0, so R = 0. Factorials grow so fast that they overpower any power of x.

Example 2: Radius of convergence equals infinity

Find the radius of convergence of

∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}

This is the series for e to the x. Apply the ratio test

L=lim⁡n→∞∣xn+1/(n+1)!xn/n!∣=lim⁡n→∞∣x∣n+1=0L = \lim_{n \to \infty} \left|\frac{x^{n+1}/(n+1)!}{x^n/n!}\right| = \lim_{n \to \infty} \frac{|x|}{n+1} = 0

The limit is 0 for every value of x, which is less than 1. The series converges for all real x, so R = infinity. The factorial in the denominator shrinks the terms fast enough to overpower any power of x.

Example 3: Finite radius with endpoint analysis

Find the interval of convergence of

∑n=1∞(−1)n(x−2)nn\sum_{n=1}^{\infty} \frac{(-1)^n(x - 2)^n}{n}

Apply the ratio test

L=lim⁡n→∞∣(x−2)n+1/(n+1)(x−2)n/n∣=lim⁡n→∞∣x−2∣⋅nn+1=∣x−2∣L = \lim_{n \to \infty} \left|\frac{(x-2)^{n+1}/(n+1)}{(x-2)^n/n}\right| = \lim_{n \to \infty} |x - 2| \cdot \frac{n}{n+1} = |x - 2|

For convergence we need L less than 1, so the distance from center must be less than 1. The radius is R = 1, and the open interval is 1 less than x less than 3.

Now check the endpoints.

At x = 1: substitute x - 2 = -1

∑n=1∞(−1)n(−1)nn=∑n=1∞1n\sum_{n=1}^{\infty} \frac{(-1)^n(-1)^n}{n} = \sum_{n=1}^{\infty} \frac{1}{n}

This is the harmonic series, which diverges.

At x = 3: substitute x - 2 = 1

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n}

This is the alternating harmonic series, which converges (conditionally).

The interval of convergence is (1, 3]. The left endpoint x = 1 is excluded (harmonic series diverges) and the right endpoint x = 3 is included (alternating harmonic converges).

Real-World Application

Section titled “Real-World Application”

Power series are everywhere in computing and science:

  • Game engines replace expensive functions (sin, cos, exp) with truncated power series for speed
  • Physics uses power series to solve differential equations that have no closed-form solution
  • Signal processing represents signals as power series (z-transforms)
  • Machine learning uses Taylor expansions to analyze and optimize loss functions
  • Computer graphics uses polynomial approximations for lighting and shading models

Quiz

Section titled “Quiz”
A power series centered at a has the form
The radius of convergence of $\sum x^n/n!$ is
The radius of convergence of $\sum n!\,x^n$ is
At the endpoints of the interval of convergence, you should
Power series are used in game engines primarily to