Practice Convergence and Divergence in Math

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

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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

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

Example 2

medium

Determine convergence of

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

Example 3

medium

Apply the ratio test to

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

Example 4

easy

The partial sums of a series are

S_n = 3 - \frac{1}{n}

. Find the sum of the series.

Example 5

challenge

Prove that the harmonic series

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

diverges by grouping.

Example 6

medium

Use limit comparison with

1/n

to determine convergence of

\sum_{n=1}^{\infty} \dfrac{n}{n^2 + 3}

Example 7

easy

True or false: if

a_n \to 0

, then

\sum a_n

converges.

Example 8

medium

Use the comparison test to determine convergence of

\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}

Example 9

hard

Find the interval of convergence of

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

Example 10

medium

Does

\sum_{n=1}^{\infty} \frac{n+1}{2n+3}

converge?

Example 11

easy

Does the harmonic series

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

converge?

Example 12

medium

Find the sum of

\sum_{n=2}^{\infty} \dfrac{1}{n^2 - 1}

via partial fractions.

Example 13

medium

Apply the ratio test to

\sum_{n=1}^{\infty} \frac{n^2}{3^n}

Example 14

medium

Use the comparison test to determine convergence of

\sum_{n=1}^{\infty} \frac{1}{n^3 + n}

Example 15

easy

Does

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

converge?

Example 16

hard

Show

\sum_{n=1}^{\infty} \dfrac{\ln n}{n^2}

converges.

Example 17

challenge

Determine whether

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

converges.

Example 18

medium

Use the ratio test to determine whether

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

converges or diverges.

Example 19

easy

If the partial sums of a series are

S_n = 4 - \dfrac{2}{n}

, what is the sum?

Example 20

easy

Does

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

converge or diverge?

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Related Concepts

Series Limit Infinite Geometric Series Taylor Series Power Series