Series Examples: 48 Problems with Answers

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: $a_1 + a_2 + a_3 + \ldots$ — 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

S_1

through

S_4

for

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

and identify the limit.

Answer

S_1=\frac{1}{2}

S_2=\frac{3}{4}

S_3=\frac{7}{8}

S_4=\frac{15}{16}

; series sum

= 1

First step

Terms:

\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots

Full solution

2
$S_1 = \frac{1}{2}$ , $S_2 = \frac{3}{4}$ , $S_3 = \frac{7}{8}$ , $S_4 = \frac{15}{16}$ .
3
Pattern: $S_n = 1 - \frac{1}{2^n} \to 1$ .
4
Alternatively, geometric series: $a = \frac{1}{2}$ , $r = \frac{1}{2}$ , sum $= \frac{a}{1-r} = 1$ .

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

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

diverges.

Example 3

medium

Find the sum

\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n

Example 4

medium

Express the repeating decimal

0.\overline{27}

as a fraction using a geometric series.

Example 5

medium

Apply the

p

-series test to

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

Example 6

medium

Use the comparison test to determine whether

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

converges.

Example 7

hard

Determine convergence of

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

and find its sum.

Example 8

hard

Find the radius of convergence of the power series

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

Example 9

challenge

Find the sum of

\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 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Example 2

medium

Use the divergence test on

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

Example 3

easy

2+4+6+8

a sequence or a series?

Example 4

easy

Find the 3rd partial sum

S_3

of the series

1+2+3+4+\cdots

Example 5

easy

Does the geometric series

1+\frac12+\frac14+\cdots

converge? To what?

Example 6

easy

Does the harmonic series

1+\frac12+\frac13+\frac14+\cdots

converge?

Example 7

easy

Use sigma notation to write

1+4+9+16

Example 8

easy

Find the sum of the finite series

\sum_{n=1}^{5} 2n

Example 9

easy

What is the difference between

a_n

and

S_n

for a series?

Example 10

easy

Does

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

(i.e.

1+2+3+\cdots

) converge?

Example 11

medium

Find the sum of the arithmetic series

\sum_{n=1}^{10}(3n+1)

Example 12

medium

Evaluate the infinite geometric series

\sum_{n=0}^{\infty} 3\left(\frac{1}{4}\right)^n

Example 13

medium

Use the formula

\sum_{n=1}^{N} n=\frac{N(N+1)}{2}

to find

\sum_{n=1}^{100} n

Example 14

medium

A series has partial sums

S_n=\frac{n}{n+1}

. What does the series converge to?

Example 15

medium

Find the sum of the telescoping series

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

Example 16

medium

Determine whether

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

converges, and if so find its sum.

Example 17

challenge

Show that

\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

\sum\frac{1}{n}

Example 19

challenge

Find the sum of

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

for

|x|<1

(hint: differentiate the geometric series).

Example 20

medium

Find the sum of the arithmetic series

\sum_{n=1}^{8}(5n-2)

Example 21

medium

Evaluate the infinite series

\sum_{n=1}^{\infty} 5\left(\frac{2}{3}\right)^{n-1}

Example 22

medium

Use

\sum_{n=1}^{N} n^2=\frac{N(N+1)(2N+1)}{6}

to find

\sum_{n=1}^{10} n^2

Example 23

easy

Find the sum

\sum_{n=1}^{4} (2n - 1)

Example 24

easy

Compute

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

Example 25

easy

Find the second partial sum

S_2

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

Example 26

easy

Does

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

converge?

Example 27

medium

Find the sum of the geometric series

\sum_{n=1}^{\infty} 5\left(\frac{1}{4}\right)^{n-1}

Example 28

medium

Apply the divergence test to

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

Example 29

medium

Find the partial sum

S_5

of the geometric series

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

Example 30

medium

Test

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

for convergence by finding

S_N

explicitly (telescoping).

Example 31

medium

Apply the

p

-series test to

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

Example 32

medium

Determine whether

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

converges.

Example 33

medium

Find the sum of the geometric series

\sum_{n=1}^{\infty} \frac{3^n}{4^{n+1}}

Example 34

hard

Apply the ratio test to

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

Example 35

hard

Find the sum of the telescoping series

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

Example 36

hard

Apply the integral test to

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

Example 37

hard

Find the sum

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

via partial fractions.

Example 38

hard

\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

\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}

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Related Concepts

Sequence Convergence and Divergence

Background Knowledge

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

sequence