Series Examples in Math

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

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

Concept 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.

Read the full concept explanation →

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 adds the terms of a sequence; an infinite one can still have a finite total if the terms shrink fast enough.

Common stuck point: The procedure for series is the easy part; the trap is concluding a series converges just because its terms go to zero. Asking "Am I adding the terms into a total, rather than just listing them by position?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I adding the terms into a total, rather than just listing them by position?

Worked Examples

Example 1

easy
Compute partial sums S1S_1 through S4S_4 for ∑n=1∞12n\sum_{n=1}^{\infty} \frac{1}{2^n} and identify the limit.

Answer

S1=12S_1=\frac{1}{2}, S2=34S_2=\frac{3}{4}, S3=78S_3=\frac{7}{8}, S4=1516S_4=\frac{15}{16}; series sum =1= 1

First step

1
Terms: 12,14,18,116,…\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots

Full solution

  1. 2
    S1=12S_1 = \frac{1}{2}, S2=34S_2 = \frac{3}{4}, S3=78S_3 = \frac{7}{8}, S4=1516S_4 = \frac{15}{16}.
  2. 3
    Pattern: Sn=1−12n→1S_n = 1 - \frac{1}{2^n} \to 1.
  3. 4
    Alternatively, geometric series: a=12a = \frac{1}{2}, r=12r = \frac{1}{2}, sum =a1−r=1= \frac{a}{1-r} = 1.
The partial sums approach 1, confirming the series converges. Each new term adds half the remaining gap to 1.

Example 2

hard
Show that the harmonic series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} diverges.

Example 3

medium
Find the sum ∑n=0∞(23)n\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n.

Example 4

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

Example 5

medium
Apply the pp-series test to ∑n=1∞1n3/2\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}.

Example 6

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

Example 7

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

Example 8

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

Example 9

challenge
Find the sum of ∑n=1∞n3n\sum_{n=1}^{\infty} \frac{n}{3^n}.

Practice Problems

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

Example 1

easy
Write the first four partial sums of 1−12+13−14+⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Example 2

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

Example 3

easy
Is 2+4+6+82+4+6+8 a sequence or a series?

Example 4

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

Example 5

easy
Does the geometric series 1+12+14+⋯1+\frac12+\frac14+\cdots converge? To what?

Example 6

easy
Does the harmonic series 1+12+13+14+⋯1+\frac12+\frac13+\frac14+\cdots converge?

Example 7

easy
Use sigma notation to write 1+4+9+161+4+9+16.

Example 8

easy
Find the sum of the finite series ∑n=152n\sum_{n=1}^{5} 2n.

Example 9

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

Example 10

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

Example 11

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

Example 12

medium
Evaluate the infinite geometric series ∑n=0∞3(14)n\sum_{n=0}^{\infty} 3\left(\frac{1}{4}\right)^n.

Example 13

medium
Use the formula ∑n=1Nn=N(N+1)2\sum_{n=1}^{N} n=\frac{N(N+1)}{2} to find ∑n=1100n\sum_{n=1}^{100} n.

Example 14

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

Example 15

medium
Find the sum of the telescoping series ∑n=1∞(1n−1n+1)\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right).

Example 16

medium
Determine whether ∑n=1∞12n\sum_{n=1}^{\infty}\frac{1}{2^n} converges, and if so find its sum.

Example 17

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

Example 18

challenge
Explain why a series can have terms approaching 0 yet still diverge, using ∑1n\sum\frac{1}{n}.

Example 19

challenge
Find the sum of ∑n=1∞nxn−1\sum_{n=1}^{\infty} n x^{n-1} for ∣x∣<1|x|<1 (hint: differentiate the geometric series).

Example 20

medium
Find the sum of the arithmetic series ∑n=18(5n−2)\sum_{n=1}^{8}(5n-2).

Example 21

medium
Evaluate the infinite series ∑n=1∞5(23)n−1\sum_{n=1}^{\infty} 5\left(\frac{2}{3}\right)^{n-1}.

Example 22

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 23

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

Example 24

easy
Compute ∑n=13n3\sum_{n=1}^{3} n^3.

Example 25

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

Example 26

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

Example 27

medium
Find the sum of the geometric series ∑n=1∞5(14)n−1\sum_{n=1}^{\infty} 5\left(\frac{1}{4}\right)^{n-1}.

Example 28

medium
Apply the divergence test to ∑n=1∞cos⁡(1n)\sum_{n=1}^{\infty} \cos\left(\frac{1}{n}\right).

Example 29

medium
Find the partial sum S5S_5 of the geometric series ∑n=1∞2⋅(1/2)n−1\sum_{n=1}^{\infty} 2 \cdot (1/2)^{n-1}.

Example 30

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 31

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

Example 32

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

Example 33

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

Example 34

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

Example 35

hard
Find the sum of the telescoping series ∑n=1∞(1n−1n+2)\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+2}\right).

Example 36

hard
Apply the integral test to ∑n=2∞1nln⁡n\sum_{n=2}^{\infty} \frac{1}{n \ln n}.

Example 37

hard
Find the sum ∑n=1∞1n(n+2)\sum_{n=1}^{\infty} \frac{1}{n(n+2)} via partial fractions.

Example 38

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

Example 39

challenge
Determine the interval of convergence for the power series ∑n=1∞(x−2)nn⋅3n\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}.
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Background Knowledge

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

sequence