Practice Infinite Geometric Series in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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).

Showing a random 20 of 50 problems.

Example 1

challenge

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

a = r

. If the sum is

\frac{1}{3}

, find

r

Example 2

challenge

Find

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

Example 3

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 4

medium

Find the sum of

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

Example 5

medium

Express

0.\overline{45}

as a fraction using an infinite geometric series.

Example 6

easy

Find the sum of

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

Example 7

easy

Find

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

Example 8

medium

A ball is dropped from 16 m and bounces to

\frac{1}{2}

its height each time. Find the total distance traveled.

Example 9

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 10

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 11

medium

If a geometric series has sum

\frac{9}{2}

and ratio

r = \frac{1}{3}

, find the first term

a

Example 12

hard

Find a closed form for

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

when

|x| < 1

Example 13

medium

A geometric series has first term

a=20

and sum

80

. Find the common ratio

r

Example 14

easy

Does

\sum_{n=0}^{\infty} (1.1)^n

converge or diverge?

Example 15

hard

Evaluate

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

Example 16

easy

Find the sum of

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

Example 17

easy

Find

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

Example 18

easy

For what values of

r

does

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

converge?

Example 19

medium

Find

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

Example 20

medium

Express

0.1\overline{6}

as a fraction (the 6 repeats).

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Related Concepts

Geometric Sequence Series Limit Convergence and Divergence Taylor Series