Convergence and Divergence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Convergence and Divergence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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?

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A series converges if its partial sums approach a finite limit, and diverges if they blow up or never settle.

Common stuck point: The procedure for convergence and divergence is the easy part; the trap is concluding convergence from terms $\to 0$ . Asking "Does the sequence of partial sums approach a single finite number as you add more terms?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the sequence of partial sums approach a single finite number as you add more terms?

Worked Examples

Example 1

medium

Use the ratio test to determine whether

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

converges or diverges.

Answer

The series converges (ratio test:

L = \frac{1}{2} < 1

First step

a_n = \frac{n}{2^n}

. Compute

L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|

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Example 2

hard

Determine whether

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

converges using the

p

-series test.

Example 3

medium

Find the sum of the geometric series

\sum_{n=0}^{\infty} \frac{2}{5^n}

Example 4

hard

Find the radius of convergence of

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

Example 5

hard

Find the interval of convergence of

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

Example 6

challenge

Prove that the harmonic series

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

diverges by grouping.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Does

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

converge or diverge?

Example 2

medium

Apply the ratio test to

\displaystyle\sum_{n=0}^{\infty} \frac{3^n}{n!}

Example 3

easy

Does the sequence

a_n = \frac{1}{n}

converge as

n \to \infty

Example 4

easy

Does

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

converge?

Example 5

easy

Does

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

converge or diverge?

Example 6

easy

Apply the

n

th-term test to

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

Example 7

easy

Does the harmonic series

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

converge?

Example 8

easy

Does

\sum_{n=1}^{\infty} \left(\frac{3}{2}\right)^n

converge?

Example 9

easy

The partial sums of a series are

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

. Find the sum of the series.

Example 10

easy

Does

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

converge?

Example 11

medium

Use the

p

-series test on

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

Example 12

medium

Use the

p

-series test on

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

Example 13

medium

Apply the ratio test to

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

Example 14

medium

Use the comparison test on

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

Example 15

medium

Apply the ratio test to

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

Example 16

medium

Determine convergence of the alternating series

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

Example 17

medium

Does

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

converge?

Example 18

medium

Use the integral test on

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

Example 19

challenge

Use the ratio test to find all

x>0

for which

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

converges.

Example 20

challenge

Determine whether

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

converges.

Example 21

medium

The ratio test gives a limit of 1 for

\sum \frac{1}{n^2}

. Explain and decide convergence.

Example 22

challenge

Show that

\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)

converges and find its sum.

Example 23

easy

Does

\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n

converge?

Example 24

easy

Does

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

converge?

Example 25

easy

Does

\sum_{n=1}^{\infty} 5^n

converge?

Example 26

easy

True or false: if

a_n \to 0

, then

\sum a_n

converges.

Example 27

medium

Apply the ratio test to

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

Example 28

medium

Apply the ratio test to

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

Example 29

medium

Use the comparison test to determine convergence of

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

Example 30

medium

Use the comparison test to determine convergence of

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

Example 31

medium

Does the alternating series

\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{\sqrt{n}}

converge?

Example 32

medium

Determine convergence of

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

Example 33

medium

Apply the root test to

\sum_{n=1}^{\infty} \left(\dfrac{n}{2n+1}\right)^n

Example 34

hard

Use the integral test on

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

Example 35

hard

Find the radius of convergence of

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

Example 36

hard

Show

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

converges.

Example 37

medium

Find the sum of

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

via partial fractions.

Example 38

medium

Use limit comparison with

1/n

to determine convergence of

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

Example 39

challenge

Determine whether

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

converges.

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Related Concepts

Series Limit Infinite Geometric Series Taylor Series Power Series

Background Knowledge

These ideas may be useful before you work through the harder examples.

serieslimitinfinite geometric series