Infinite Geometric Series Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Infinite Geometric 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 sum of all terms of a geometric sequence with common ratio $|r| < 1$ . The infinite sum converges to $\frac{a}{1-r}$ , where $a$ is the first term.

If each term is a fixed fraction of the previous one, the terms shrink fast enough that the total sum stays finite. Imagine walking halfway to a wall, then half the remaining distance, then half again—you approach the wall but the total distance is finite (exactly the full distance to the wall).

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: When each term is a fixed fraction $|r|<1$ of the last, the shrinking terms add to a finite sum $\frac{a}{1-r}$ .

Common stuck point: The procedure for infinite geometric series is the easy part; the trap is using $\frac{a}{1-r}$ without checking $|r|<1$ . Asking "Are the terms a geometric sequence with $|r|<1$ , and am I asked for the sum of all of them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the terms a geometric sequence with $|r|<1$ , and am I asked for the sum of all of them?

Worked Examples

Example 1

easy

Find the sum of the infinite geometric series

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

Answer

\frac{3}{2}

First step

Identify: first term

a = 1

(when

n=0

), common ratio

r = \frac{1}{3}

Full solution

2
Check: $|r| = \frac{1}{3} < 1$ , so the series converges.
3
Sum formula: $S = \frac{a}{1-r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$ .

Always verify

|r| < 1

before applying the formula. The first term

a

is the value at

n=0

, which is

\left(\frac{1}{3}\right)^0 = 1

Example 2

medium

Convert the repeating decimal

0.\overline{27}

to a fraction using an infinite geometric series.

Example 3

medium

Find the sum of the infinite series

12 + 4 + \frac{4}{3} + \frac{4}{9} + \cdots

Example 4

medium

Express

0.\overline{45}

as a fraction using an infinite geometric series.

Example 5

medium

Express

0.\overline{142857}

as a fraction using an infinite geometric series.

Example 6

medium

Express

0.1\overline{6}

as a fraction (the 6 repeats).

Example 7

hard

Find a closed form for

\sum_{n=0}^{\infty} (n+1) x^n

when

|x| < 1

Practice Problems

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

Example 1

easy

Find the sum:

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

Example 2

hard

For what values of

x

does

\displaystyle\sum_{n=0}^{\infty} (x-1)^n

converge, and what is the sum?

Example 3

easy

Find the sum of

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots

Example 4

easy

Find

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

Example 5

easy

Does

1 + 2 + 4 + 8 + \cdots

converge?

Example 6

easy

Find the sum of

9 + 3 + 1 + \frac{1}{3} + \cdots

Example 7

easy

What is the common ratio of

\frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \cdots

Example 8

easy

Find

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

Example 9

easy

Express

0.\overline{3} = 0.333\ldots

as an infinite geometric series sum.

Example 10

easy

For what values of

r

does

\sum_{n=0}^{\infty} r^n

converge?

Example 11

medium

Find the sum of

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

Example 12

medium

A ball is dropped from 16 m and bounces to

\frac{1}{2}

its height each time. Find the total distance traveled.

Example 13

medium

The sum of an infinite geometric series is 12 and its first term is 4. Find

r

Example 14

medium

Find

\sum_{n=0}^{\infty} \frac{(-1)^n}{2^n}

Example 15

medium

Convert

0.\overline{27} = 0.272727\ldots

to a fraction using a geometric series.

Example 16

medium

\sum_{n=1}^{\infty} a r^{n-1} = 10

with

a = 6

, find

r

Example 17

medium

Find the sum

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

Example 18

medium

An infinite series has

S = \frac{a}{1-r} = 50

and

r = 0.8

. Find the first term

a

Example 19

challenge

Find

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

Example 20

challenge

A square has side 1. A smaller square is inscribed by joining midpoints, repeated infinitely. Find the total area of all squares.

Example 21

challenge

Find all

x

for which

\sum_{n=0}^{\infty} (x-2)^n

converges, and give the sum.

Example 22

medium

A drug dose of 100 mg is taken daily; each day

20\%

remains from prior doses. Find the long-run amount just after a dose.

Example 23

easy

Find the sum of the infinite series

6 + 2 + \frac{2}{3} + \frac{2}{9} + \cdots

Example 24

easy

Find the sum of

\sum_{n=0}^{\infty} \left(\frac{2}{7}\right)^n

Example 25

easy

Find the sum:

10 - 5 + \frac{5}{2} - \frac{5}{4} + \cdots

Example 26

easy

Find the sum of

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

Example 27

medium

Find the sum of

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

Example 28

medium

A geometric series has first term

a=20

and sum

80

. Find the common ratio

r

Example 29

medium

Find the sum:

\sum_{n=1}^{\infty} \frac{4}{5^n}

Example 30

medium

If a geometric series has sum

\frac{9}{2}

and ratio

r = \frac{1}{3}

, find the first term

a

Example 31

medium

Find the sum:

\sum_{n=0}^{\infty} \frac{(-2)^n}{3^{n+1}}

Example 32

medium

A pendulum swings back, traveling

40

cm. Each next swing covers

80\%

of the previous. Find the total distance.

Example 33

hard

For what values of

x

does

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

converge? Give the sum on the interval of convergence.

Example 34

hard

A ball is dropped from

20

m and rebounds to

\frac{3}{5}

of its height each bounce. Find the total vertical distance until it comes to rest.

Example 35

hard

Find the sum:

\sum_{n=0}^{\infty} (\sin^2 \theta)^n

assuming

\sin \theta \neq \pm 1

Example 36

hard

A patient takes

200

mg of medication every

6

hours;

25\%

remains in the body when the next dose is taken. Find the long-run amount just after a dose.

Example 37

hard

An equilateral triangle has side

6

. The midpoints form a smaller equilateral triangle, and this is repeated forever. Find the total perimeter of all triangles.

Example 38

hard

Evaluate

\sum_{n=0}^{\infty} \frac{n+1}{3^n}

Example 39

challenge

The first term of a geometric series equals its common ratio:

a = r

. If the sum is

\frac{1}{3}

, find

r

Example 40

challenge

Evaluate

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

← Back to Infinite Geometric Series Formula Explained Practice Mode

Related Concepts

Geometric Sequence Series Limit Convergence and Divergence Taylor Series

Background Knowledge

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

geometric sequenceserieslimit