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 →0\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 ∑n=1∞n2n\displaystyle\sum_{n=1}^{\infty} \frac{n}{2^n} converges or diverges.

Answer

The series converges (ratio test: L=12<1L = \frac{1}{2} < 1).

First step

1
an=n2na_n = \frac{n}{2^n}. Compute L=lim⁡n→∞∣an+1an∣L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.

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Example 2

hard
Determine whether ∑n=1∞1n2\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} converges using the pp-series test.

Example 3

medium
Find the sum of the geometric series ∑n=0∞25n\sum_{n=0}^{\infty} \frac{2}{5^n}.

Example 4

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

Example 5

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

Example 6

challenge
Prove that the harmonic series ∑n=1∞1n\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 ∑n=1∞1n1/2\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1/2}} converge or diverge?

Example 2

medium
Apply the ratio test to ∑n=0∞3nn!\displaystyle\sum_{n=0}^{\infty} \frac{3^n}{n!}.

Example 3

easy
Does the sequence an=1na_n = \frac{1}{n} converge as n→∞n \to \infty?

Example 4

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

Example 5

easy
Does ∑n=1∞n\sum_{n=1}^{\infty} n converge or diverge?

Example 6

easy
Apply the nnth-term test to ∑n=1∞nn+1\sum_{n=1}^{\infty} \frac{n}{n+1}.

Example 7

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

Example 8

easy
Does ∑n=1∞(32)n\sum_{n=1}^{\infty} \left(\frac{3}{2}\right)^n converge?

Example 9

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

Example 10

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

Example 11

medium
Use the pp-series test on ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}.

Example 12

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

Example 13

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

Example 14

medium
Use the comparison test on ∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}.

Example 15

medium
Apply the ratio test to ∑n=1∞n2n\sum_{n=1}^{\infty} \frac{n}{2^n}.

Example 16

medium
Determine convergence of the alternating series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}.

Example 17

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

Example 18

medium
Use the integral test on ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}.

Example 19

challenge
Use the ratio test to find all x>0x>0 for which ∑n=1∞xnn\sum_{n=1}^{\infty} \frac{x^n}{n} converges.

Example 20

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

Example 21

medium
The ratio test gives a limit of 1 for ∑1n2\sum \frac{1}{n^2}. Explain and decide convergence.

Example 22

challenge
Show that ∑n=1∞(1n−1n+1)\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) converges and find its sum.

Example 23

easy
Does ∑n=1∞(13)n\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n converge?

Example 24

easy
Does ∑n=1∞1n3\sum_{n=1}^{\infty} \frac{1}{n^3} converge?

Example 25

easy
Does ∑n=1∞5n\sum_{n=1}^{\infty} 5^n converge?

Example 26

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

Example 27

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

Example 28

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

Example 29

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

Example 30

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

Example 31

medium
Does the alternating series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{\sqrt{n}} converge?

Example 32

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

Example 33

medium
Apply the root test to ∑n=1∞(n2n+1)n\sum_{n=1}^{\infty} \left(\dfrac{n}{2n+1}\right)^n.

Example 34

hard
Use the integral test on ∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}.

Example 35

hard
Find the radius of convergence of ∑n=1∞xnn2\sum_{n=1}^{\infty} \dfrac{x^n}{n^2}.

Example 36

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

Example 37

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

Example 38

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 39

challenge
Determine whether ∑n=2∞1n(ln⁡n)2\sum_{n=2}^{\infty} \dfrac{1}{n (\ln n)^2} converges.
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Background Knowledge

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

serieslimitinfinite geometric series