Practice Convergence and Divergence 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

A series converges if the sequence of its partial sums approaches a finite limit. A series diverges if the partial sums grow without bound or oscillate without settling.

Convergence means the infinite sum adds up to a finite number—each new term adds less and less, and the total stabilizes. Divergence means the sum either blows up to infinity or never settles down. The key question: does adding infinitely many terms produce a finite result?

Showing a random 20 of 50 problems.

Example 1

medium
Apply the ratio test to ∑n=1∞n!2n\sum_{n=1}^{\infty} \frac{n!}{2^n}.

Example 2

medium
Determine convergence of ∑n=1∞(−1)nn2\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}.

Example 3

medium
Apply the ratio test to ∑n=1∞2nn!\sum_{n=1}^{\infty} \frac{2^n}{n!}.

Example 4

easy
The partial sums of a series are Sn=3−1nS_n = 3 - \frac{1}{n}. Find the sum of the series.

Example 5

challenge
Prove that the harmonic series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} diverges by grouping.

Example 6

medium
Use limit comparison with 1/n1/n to determine convergence of ∑n=1∞nn2+3\sum_{n=1}^{\infty} \dfrac{n}{n^2 + 3}.

Example 7

easy
True or false: if an→0a_n \to 0, then ∑an\sum a_n converges.

Example 8

medium
Use the comparison test to determine convergence of ∑n=2∞1n−1\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}.

Example 9

hard
Find the interval of convergence of ∑n=1∞(x−2)nn\sum_{n=1}^{\infty} \dfrac{(x-2)^n}{n}.

Example 10

medium
Does ∑n=1∞n+12n+3\sum_{n=1}^{\infty} \frac{n+1}{2n+3} converge?

Example 11

easy
Does the harmonic series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} converge?

Example 12

medium
Find the sum of ∑n=2∞1n2−1\sum_{n=2}^{\infty} \dfrac{1}{n^2 - 1} via partial fractions.

Example 13

medium
Apply the ratio test to ∑n=1∞n23n\sum_{n=1}^{\infty} \frac{n^2}{3^n}.

Example 14

medium
Use the comparison test to determine convergence of ∑n=1∞1n3+n\sum_{n=1}^{\infty} \frac{1}{n^3 + n}.

Example 15

easy
Does ∑n=1∞12n\sum_{n=1}^{\infty} \frac{1}{2^n} converge?

Example 16

hard
Show ∑n=1∞ln⁡nn2\sum_{n=1}^{\infty} \dfrac{\ln n}{n^2} converges.

Example 17

challenge
Determine whether ∑n=2∞1nln⁡n\sum_{n=2}^{\infty} \frac{1}{n \ln n} converges.

Example 18

medium
Use the ratio test to determine whether ∑n=1∞n2n\displaystyle\sum_{n=1}^{\infty} \frac{n}{2^n} converges or diverges.

Example 19

easy
If the partial sums of a series are Sn=4−2nS_n = 4 - \dfrac{2}{n}, what is the sum?

Example 20

easy
Does ∑n=1∞n\sum_{n=1}^{\infty} n converge or diverge?
← Back to Convergence and Divergence Worked Examples Formula Explained