Power Series Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Power Series.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An infinite series of the form

\sum_{n = 0}^{\infty} a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + \dots

where

c

is the center and

a_n

are the coefficients. A power series defines a function of

x

wherever it converges.

A power series is an 'infinite polynomial' centered at $c$ . For each value of $x$ , you get a number series that may or may not converge. The set of $x$ -values where it converges forms an interval centered at $c$ , and within that interval, the power series behaves like a well-defined function.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A power series $\sum a_n(x-c)^n$ defines a function on the interval of $x$ where it converges.

Common stuck point: The procedure for power series is the easy part; the trap is reporting only the radius and skipping endpoints. Asking "Is this a series whose terms are coefficients times powers of $(x-c)$ , with convergence depending on the value of $x$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a series whose terms are coefficients times powers of $(x-c)$ , with convergence depending on the value of $x$ ?

Worked Examples

Example 1

medium

Find the radius of convergence of

\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}

Answer

R = 1

First step

Apply the ratio test: compute

\left|\frac{a_{n+1}}{a_n}\right|

where

a_n = \frac{x^n}{n+1}

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Example 2

hard

Find the interval of convergence of

\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n}

Example 3

medium

Find the interval of convergence of

\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}

Example 4

medium

Find the sum of

\sum_{n=0}^{\infty} \frac{(-1)^n}{2^n}

Example 5

medium

Express

\frac{1}{(1-x)^2}

as a power series.

Example 6

hard

Find the interval of convergence of

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-3)^n}{n \cdot 2^n}

Example 7

hard

Show

\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)} = \frac{\pi}{4}

using a power-series identity.

Example 8

hard

Find the power series of

f(x) = \frac{1}{2 - x}

centered at

0

. Give the radius of convergence.

Example 9

hard

Show that

\sum_{n=0}^{\infty} x^{n!}

has radius of convergence

1

Example 10

challenge

Find the radius of convergence of

\sum_{n=1}^{\infty} \frac{x^n}{n^{\ln n}}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the radius of convergence of

\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}

Example 2

medium

Differentiate

\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

term by term to find a new series identity.

Example 3

easy

What is the center of

\sum_{n=0}^{\infty}a_n(x-3)^n

Example 4

easy

Find the radius of convergence of

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

Example 5

easy

Find the radius of convergence of

\sum_{n=0}^{\infty}n!\,x^n

Example 6

easy

Find the radius of convergence of

\sum_{n=0}^{\infty}x^n

Example 7

easy

For a power series with

R=2

centered at

0

, give the open interval of convergence.

Example 8

easy

Why must endpoints of the interval of convergence be checked separately?

Example 9

easy

Does term-by-term differentiation change the radius of convergence?

Example 10

easy

What function does

\sum_{n=0}^{\infty}x^n

equal for

|x|<1

Example 11

medium

Find the radius of convergence of

\sum_{n=0}^{\infty}\frac{x^n}{2^n}

Example 12

medium

Find the interval of convergence of

\sum_{n=1}^{\infty}\frac{x^n}{n}

Example 13

medium

Find the radius of convergence of

\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n}

Example 14

medium

Find the interval of convergence of

\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n}

Example 15

medium

Differentiate

\frac{1}{1-x}=\sum x^n

to find a series for

\frac{1}{(1-x)^2}

Example 16

medium

Integrate

\frac{1}{1+x}=\sum(-1)^n x^n

to find a series for

\ln(1+x)

Example 17

medium

Find the radius of convergence of

\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}

Example 18

medium

Find the radius of convergence of

\sum_{n=1}^{\infty}\frac{n}{2^n}x^n

Example 19

medium

Find the interval of convergence of

\sum_{n=1}^{\infty}\frac{x^n}{n^2}

Example 20

challenge

Find the full interval of convergence of

\sum_{n=1}^{\infty}\frac{(x-1)^n}{n\,2^n}

Example 21

challenge

Use

\frac{1}{1-x}=\sum x^n

to find a power series for

\frac{x}{1-x^2}

and its radius.

Example 22

challenge

Find the radius of convergence of

\sum_{n=0}^{\infty}\frac{(2x)^n}{n^2+1}

Example 23

easy

Find the radius of convergence of

\sum_{n=0}^{\infty} \frac{x^n}{2^n}

Example 24

easy

What does

\sum_{n=0}^{\infty} (-1)^n x^n

equal for

|x| < 1

Example 25

easy

Find the interval of convergence of

\sum_{n=1}^{\infty} \frac{x^n}{n^2}

Example 26

easy

Find the radius of convergence of

\sum_{n=0}^{\infty} \frac{x^n}{n+1}

Example 27

easy

Find the radius of convergence of

\sum_{n=0}^{\infty} (3x)^n

Example 28

medium

For what

x

does

\sum_{n=0}^{\infty} \frac{(x-1)^n}{4^n}

converge?

Example 29

medium

Integrate

\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}

term by term to get a series for

-\ln(1-x)

Example 30

medium

Find the radius of convergence of

\sum_{n=0}^{\infty} \frac{n!\, x^n}{n^n}

Example 31

medium

Find the radius of convergence of

\sum_{n=0}^{\infty} \frac{(x-4)^n}{\sqrt{n+1}}

Example 32

medium

What function is represented by

\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}

Example 33

hard

What is the radius of convergence of

\sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^2} x^n

Example 34

hard

Use a power-series representation to compute

\int_0^{0.5} \frac{1}{1 + x^4}\,dx

to 3 decimal places.

Example 35

hard

Find the power series for

\frac{x}{(1 - x)^2}

Example 36

hard

Find a closed form for

\sum_{n=0}^{\infty} (n+1) x^n

when

|x| < 1

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Related Concepts

Convergence and Divergence Taylor Series Sigma Notation

Background Knowledge

These ideas may be useful before you work through the harder examples.

convergence divergencetaylor seriessigma notation