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 ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+⋯\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots where cc is the center and ana_n are the coefficients. A power series defines a function of xx wherever it converges.

A power series is an 'infinite polynomial' centered at cc. For each value of xx, you get a number series that may or may not converge. The set of xx-values where it converges forms an interval centered at cc, 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 ∑n=0∞xn\sum_{n=0}^{\infty}x^n equal for ∣x∣<1|x|<1?

Example 2

challenge
Use 11−x=∑xn\frac{1}{1-x}=\sum x^n to find a power series for x1−x2\frac{x}{1-x^2} and its radius.

Example 3

medium
Differentiate ∑n=0∞xn=11−x\sum_{n=0}^{\infty} x^n = \frac{1}{1-x} term by term to find a new series identity.

Example 4

easy
What does ∑n=0∞(−1)nxn\sum_{n=0}^{\infty} (-1)^n x^n equal for ∣x∣<1|x| < 1?

Example 5

medium
For what xx does ∑n=0∞(x−1)n4n\sum_{n=0}^{\infty} \frac{(x-1)^n}{4^n} converge?

Example 6

medium
Find the radius of convergence of ∑n=0∞x2nn!\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}.

Example 7

challenge
Find the radius of convergence of ∑n=0∞(2x)nn2+1\sum_{n=0}^{\infty}\frac{(2x)^n}{n^2+1}.

Example 8

hard
If ∑anxn\sum a_n x^n has R=5R = 5 and ∑bnxn\sum b_n x^n has R=3R = 3, what can you say about RR for ∑(an+bn)xn\sum (a_n + b_n) x^n?

Example 9

medium
Find the radius of convergence of ∑n=0∞xnn+1\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}.

Example 10

medium
Find the interval of convergence of ∑n=0∞(x−2)n3n\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n}.

Example 11

medium
Express 1(1−x)2\frac{1}{(1-x)^2} as a power series.

Example 12

hard
Show that ∑n=0∞xn!\sum_{n=0}^{\infty} x^{n!} has radius of convergence 11.

Example 13

easy
True or false: the radius of convergence of a power series can be 00.

Example 14

easy
What is the center of ∑n=0∞an(x+5)n\sum_{n=0}^{\infty} a_n (x+5)^n?

Example 15

easy
Find the radius of convergence of ∑n=0∞xn2n\sum_{n=0}^{\infty} \frac{x^n}{2^n}.

Example 16

hard
Find the interval of convergence of ∑n=1∞(−1)n+1(x−3)nn⋅2n\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)=12−xf(x) = \frac{1}{2 - x} centered at 00. Give the radius of convergence.

Example 18

easy
For a power series with R=2R=2 centered at 00, give the open interval of convergence.

Example 19

medium
Find the sum of ∑n=0∞(−1)n2n\sum_{n=0}^{\infty} \frac{(-1)^n}{2^n}.

Example 20

easy
Find the radius of convergence of ∑n=0∞xnn!\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}.
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