Infinite geometric series worksheet for Algebra 2

When the ratio is between -1 and 1 the terms shrink fast enough that the total settles on a number. One formula: the first term over one minus the ratio.

Also taught in Precalculus, at its own level and with its own examples.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Infinite Geometric Series Date Period Find each sum or say that it diverges, or convert as asked. Sum an infinite geometric series Find the sum: $- 5 - \frac{5}{3} - \frac{5}{9} - \dots$ $\sum_{k = 0}^{\infty} 9 ⁣ {(\frac{1}{3})}^{k}$ Find the sum: $8 + \frac{8}{3} + \frac{8}{9} + \dots$ $\sum_{k = 1}^{\infty} 4 ⁣ {(\frac{1}{5})}^{k - 1}$ Find the sum: $- 4 - 1 - \frac{1}{4} - \dots$ $\sum_{k = 0}^{\infty} 8 ⁣ {(\frac{1}{2})}^{k}$ Find the sum: $3 + 1 + \frac{1}{3} + \dots$ $\sum_{k = 1}^{\infty} 11 ⁣ {(\frac{1}{4})}^{k - 1}$ Find the sum: $4 + 2 + 1 + \dots$ $\sum_{k = 1}^{\infty} 5 ⁣ {(\frac{2}{3})}^{k - 1}$ Decide whether each series converges, and find its sum if it does Does $64 + 16 + 4 + \dots$ converge? If so, find its sum. Does $48 - 36 + 27 + \dots$ converge? If so, find its sum. Does $4 + 6 + 9 + \dots$ converge? If so, find its sum. Does $9 - 3 + 1 + \dots$ converge? If so, find its sum. Does $32 + 40 + 50 + \dots$ converge? If so, find its sum. Convert between fractions and repeating decimals Write $0. \overline{8}$ as a fraction in lowest terms Write $\frac{5}{6}$ as a decimal Write $0.08 \overline{3}$ as a fraction in lowest terms Write $\frac{1}{6}$ as a decimal Write $\frac{2}{3}$ as a decimal Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Infinite Geometric Series

Date Period

Find each sum or say that it diverges, or convert as asked.

Sum an infinite geometric series
Find the sum:
$- 5 - \frac{5}{3} - \frac{5}{9} - \dots$
$\sum_{k = 0}^{\infty} 9 ⁣ {(\frac{1}{3})}^{k}$
Find the sum:
$8 + \frac{8}{3} + \frac{8}{9} + \dots$
$\sum_{k = 1}^{\infty} 4 ⁣ {(\frac{1}{5})}^{k - 1}$
Find the sum:
$- 4 - 1 - \frac{1}{4} - \dots$
$\sum_{k = 0}^{\infty} 8 ⁣ {(\frac{1}{2})}^{k}$
Find the sum:
$3 + 1 + \frac{1}{3} + \dots$
$\sum_{k = 1}^{\infty} 11 ⁣ {(\frac{1}{4})}^{k - 1}$
Find the sum: $4 + 2 + 1 + \dots$
$\sum_{k = 1}^{\infty} 5 ⁣ {(\frac{2}{3})}^{k - 1}$

Decide whether each series converges, and find its sum if it does
Does $64 + 16 + 4 + \dots$ converge? If so, find its sum.
Does $48 - 36 + 27 + \dots$ converge? If so, find its sum.
Does $4 + 6 + 9 + \dots$ converge? If so, find its sum.
Does $9 - 3 + 1 + \dots$ converge? If so, find its sum.
Does $32 + 40 + 50 + \dots$ converge? If so, find its sum.

Convert between fractions and repeating decimals
Write $0. \overline{8}$ as a fraction in lowest terms
Write $\frac{5}{6}$ as a decimal
Write $0.08 \overline{3}$ as a fraction in lowest terms
Write $\frac{1}{6}$ as a decimal
Write $\frac{2}{3}$ as a decimal



Sigma Prep · sigmaprep.io/worksheets Name Infinite Geometric SeriesAnswer keyVersion 1 Date Period Find each sum or say that it diverges, or convert as asked. Sum an infinite geometric series Find the sum: $- 5 - \frac{5}{3} - \frac{5}{9} - \dots$ $- \frac{15}{2}$ $\sum_{k = 0}^{\infty} 9 ⁣ {(\frac{1}{3})}^{k}$ $\frac{27}{2}$ Find the sum: $8 + \frac{8}{3} + \frac{8}{9} + \dots$ $12$ $\sum_{k = 1}^{\infty} 4 ⁣ {(\frac{1}{5})}^{k - 1}$ $5$ Find the sum: $- 4 - 1 - \frac{1}{4} - \dots$ $- \frac{16}{3}$ $\sum_{k = 0}^{\infty} 8 ⁣ {(\frac{1}{2})}^{k}$ $16$ Find the sum: $3 + 1 + \frac{1}{3} + \dots$ $\frac{9}{2}$ $\sum_{k = 1}^{\infty} 11 ⁣ {(\frac{1}{4})}^{k - 1}$ $\frac{44}{3}$ Find the sum: $4 + 2 + 1 + \dots$ $8$ $\sum_{k = 1}^{\infty} 5 ⁣ {(\frac{2}{3})}^{k - 1}$ $15$ Decide whether each series converges, and find its sum if it does Does $64 + 16 + 4 + \dots$ converge? If so, find its sum.Converges to $\frac{256}{3}$ Does $48 - 36 + 27 + \dots$ converge? If so, find its sum.Converges to $\frac{192}{7}$ Does $4 + 6 + 9 + \dots$ converge? If so, find its sum.Diverges Does $9 - 3 + 1 + \dots$ converge? If so, find its sum.Converges to $\frac{27}{4}$ Does $32 + 40 + 50 + \dots$ converge? If so, find its sum.Diverges Convert between fractions and repeating decimals Write $0. \overline{8}$ as a fraction in lowest terms $\frac{8}{9}$ Write $\frac{5}{6}$ as a decimal $0.8 \overline{3}$ Write $0.08 \overline{3}$ as a fraction in lowest terms $\frac{1}{12}$ Write $\frac{1}{6}$ as a decimal $0.1 \overline{6}$ Write $\frac{2}{3}$ as a decimal $0. \overline{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Infinite Geometric SeriesAnswer keyVersion 1

Date Period

Find each sum or say that it diverges, or convert as asked.

Sum an infinite geometric series
Find the sum:
$- 5 - \frac{5}{3} - \frac{5}{9} - \dots$ $- \frac{15}{2}$
$\sum_{k = 0}^{\infty} 9 ⁣ {(\frac{1}{3})}^{k}$ $\frac{27}{2}$
Find the sum:
$8 + \frac{8}{3} + \frac{8}{9} + \dots$ $12$
$\sum_{k = 1}^{\infty} 4 ⁣ {(\frac{1}{5})}^{k - 1}$ $5$
Find the sum:
$- 4 - 1 - \frac{1}{4} - \dots$ $- \frac{16}{3}$
$\sum_{k = 0}^{\infty} 8 ⁣ {(\frac{1}{2})}^{k}$ $16$
Find the sum:
$3 + 1 + \frac{1}{3} + \dots$ $\frac{9}{2}$
$\sum_{k = 1}^{\infty} 11 ⁣ {(\frac{1}{4})}^{k - 1}$ $\frac{44}{3}$
Find the sum: $4 + 2 + 1 + \dots$ $8$
$\sum_{k = 1}^{\infty} 5 ⁣ {(\frac{2}{3})}^{k - 1}$ $15$

Decide whether each series converges, and find its sum if it does
Does $64 + 16 + 4 + \dots$ converge? If so, find its sum.Converges to $\frac{256}{3}$
Does $48 - 36 + 27 + \dots$ converge? If so, find its sum.Converges to $\frac{192}{7}$
Does $4 + 6 + 9 + \dots$ converge? If so, find its sum.Diverges
Does $9 - 3 + 1 + \dots$ converge? If so, find its sum.Converges to $\frac{27}{4}$
Does $32 + 40 + 50 + \dots$ converge? If so, find its sum.Diverges

Convert between fractions and repeating decimals
Write $0. \overline{8}$ as a fraction in lowest terms $\frac{8}{9}$
Write $\frac{5}{6}$ as a decimal $0.8 \overline{3}$
Write $0.08 \overline{3}$ as a fraction in lowest terms $\frac{1}{12}$
Write $\frac{1}{6}$ as a decimal $0.1 \overline{6}$
Write $\frac{2}{3}$ as a decimal $0. \overline{6}$

How to do these

Find the sum: $12, 6, 3, \dots$

Each term is half the one before, so r is one half.
The sum is 12 over 1 minus one half.

The answer is $24$ .

Where students go wrong

Using the formula when the ratio is bigger than 1. A series like 2, 6, 18 grows without limit and has no sum at all, and the formula will happily give a negative number for it.

Also called sum to infinity or convergent series.

What you can put on this worksheet

The sum of an infinite geometric series: Find the sum:
$7 + \frac{7}{3} + \frac{7}{9} + \dots$ → $\frac{21}{2}$
Converge or diverge: Does $27 + 36 + 48 + \dots$ converge? If so, find its sum. → Diverges
Written in sigma notation: $\sum_{k = 0}^{\infty} 10 ⁣ {(\frac{1}{4})}^{k}$ → $\frac{40}{3}$
Repeating decimals: Write $\frac{5}{9}$ as a decimal → $0. \overline{5}$
Sigma that may diverge, decimals, finding the ratio: Find the sum of the infinite series: $a_{1} = 3.5, r = 0.75$ → $S = 14$

Questions about these worksheets

Yes. Every infinite geometric series sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the infinite geometric series questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: the sum of an infinite geometric series, converge or diverge, written in sigma notation, repeating decimals and sigma that may diverge, decimals, finding the ratio. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the formula when the ratio is bigger than 1. A series like 2, 6, 18 grows without limit and has no sum at all, and the formula will happily give a negative number for it.

Infinite geometric series worksheet for Algebra 2

Sum an infinite geometric series

Decide whether each series converges, and find its sum if it does

Convert between fractions and repeating decimals

Sum an infinite geometric series

Decide whether each series converges, and find its sum if it does

Convert between fractions and repeating decimals

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these infinite geometric series worksheets free?+

Do the infinite geometric series worksheets come with an answer key?+

Are these infinite geometric series worksheets printable?+

What grade are these infinite geometric series worksheets for?+

Can I get a different version of the same infinite geometric series worksheet?+

Is this a free alternative to Kuta Software for infinite geometric series?+

Can I choose how hard the infinite geometric series questions are?+

What kinds of question are on a infinite geometric series worksheet?+

What do students get wrong with infinite geometric series?+