Practice Series in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The result of adding all the terms of a sequence together, either finitely or infinitely many terms.

Add up all the terms: a1+a2+a3+…a_1 + a_2 + a_3 + \ldots — an infinite series can still have a finite sum if terms shrink fast enough.

Showing a random 20 of 50 problems.

Example 1

medium
Find the sum of the arithmetic series ∑n=110(3n+1)\sum_{n=1}^{10}(3n+1).

Example 2

easy
Does ∑n=1∞n\sum_{n=1}^{\infty} n (i.e. 1+2+3+⋯1+2+3+\cdots) converge?

Example 3

medium
Determine whether ∑n=1∞(−1)n+11n\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} converges.

Example 4

medium
Apply the pp-series test to ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}.

Example 5

easy
What is the difference between ana_n and SnS_n for a series?

Example 6

easy
Find the sum of the infinite geometric series 1+13+19+⋯1 + \frac{1}{3} + \frac{1}{9} + \cdots.

Example 7

easy
What is sigma notation for 11+12+13+⋯\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots?

Example 8

medium
Use ∑n=1Nn2=N(N+1)(2N+1)6\sum_{n=1}^{N} n^2=\frac{N(N+1)(2N+1)}{6} to find ∑n=110n2\sum_{n=1}^{10} n^2.

Example 9

medium
Use the divergence test on ∑n=1∞n2n+1\sum_{n=1}^{\infty} \frac{n}{2n+1}.

Example 10

easy
Find the 3rd partial sum S3S_3 of the series 1+2+3+4+⋯1+2+3+4+\cdots.

Example 11

easy
Find the sum ∑n=14(2n−1)\sum_{n=1}^{4} (2n - 1).

Example 12

medium
Test ∑n=1∞1n(n+1)\sum_{n=1}^{\infty} \frac{1}{n(n+1)} for convergence by finding SNS_N explicitly (telescoping).

Example 13

hard
Is ∑n=1∞n2n3+1\sum_{n=1}^{\infty} \frac{n^2}{n^3 + 1} convergent or divergent?

Example 14

hard
Determine convergence of ∑n=1∞n2n\sum_{n=1}^{\infty} \frac{n}{2^n} and find its sum.

Example 15

medium
Find the sum of the geometric series ∑n=1∞3n4n+1\sum_{n=1}^{\infty} \frac{3^n}{4^{n+1}}.

Example 16

medium
A series has partial sums Sn=nn+1S_n=\frac{n}{n+1}. What does the series converge to?

Example 17

easy
Does ∑n=1∞(−1)n\sum_{n=1}^{\infty} (-1)^n converge?

Example 18

easy
Find the second partial sum S2S_2 of ∑n=1∞1/n2\sum_{n=1}^{\infty} 1/n^2.

Example 19

medium
Express the repeating decimal 0.27‾0.\overline{27} as a fraction using a geometric series.

Example 20

easy
Does the geometric series 1+12+14+⋯1+\frac12+\frac14+\cdots converge? To what?
← Back to Series Worked Examples Formula Explained