Practice Power Series in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

What function does

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

equal for

|x|<1

Example 2

challenge

Use

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

to find a power series for

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

and its radius.

Example 3

medium

Differentiate

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

term by term to find a new series identity.

Example 4

easy

What does

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

equal for

|x| < 1

Example 5

medium

For what

x

does

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

converge?

Example 6

medium

Find the radius of convergence of

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

Example 7

challenge

Find the radius of convergence of

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

Example 8

hard

\sum a_n x^n

has

R = 5

and

\sum b_n x^n

has

R = 3

, what can you say about

R

for

\sum (a_n + b_n) x^n

Example 9

medium

Find the radius of convergence of

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

Example 10

medium

Find the interval of convergence of

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

Example 11

medium

Express

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

as a power series.

Example 12

hard

Show that

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

has radius of convergence

1

Example 13

easy

True or false: the radius of convergence of a power series can be

0

Example 14

easy

What is the center of

\sum_{n=0}^{\infty} a_n (x+5)^n

Example 15

easy

Find the radius of convergence of

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

Example 16

hard

Find the interval of convergence of

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

Example 17

hard

Find the power series of

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

centered at

0

. Give the radius of convergence.

Example 18

easy

For a power series with

R=2

centered at

0

, give the open interval of convergence.

Example 19

medium

Find the sum of

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

Example 20

easy

Find the radius of convergence of

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

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Related Concepts

Convergence and Divergence Taylor Series Sigma Notation