Infinite Geometric Sum Calculator
An infinite geometric series adds up a first term and then each later term multiplied by a fixed common ratio, going on forever. When the common ratio lies strictly between minus one and one, the terms shrink fast enough that the endless sum settles on a finite value, and this calculator finds it. It takes the first term and the common ratio and returns the sum to infinity using the standard closed-form result that the sum equals the first term divided by one minus the ratio. The condition for convergence matters: if the absolute value of the ratio is one or larger, the terms do not shrink and the series has no finite sum, so the formula does not apply. Within the convergent range the result is exact, no matter how many terms you imagine. Mathematicians, physicists, economists and students meet this series in everything from repeating decimals and present-value calculations to fractal lengths and probability. Enter the first term and ratio to get the sum immediately, with the ratio left editable so you can explore convergence. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A convergent geometric series sums to a / (1 - r) when the ratio is between -1 and 1. For first term a = 10 and ratio r = 0.5, the sum is 20.00.
Infinite Geometric Sum formula
S = a / (1 - r) for |r| < 1
S = sum to infinity
a = first term
r = common ratio
valid only when -1 < r < 1
Each term is the previous term times the ratio. When the ratio's absolute value is below one, the terms shrink to zero and the endless sum converges to a over one minus r. Otherwise there is no finite sum.
Worked example
Find the sum to infinity of a geometric series with first term 10 and common ratio 0.5.
- Check convergence: |0.5| < 1, so the series converges
- Denominator = 1 - 0.5 = 0.5
- S = 10 / 0.5 = 20.00
The sum to infinity is 20.00. These are the calculator's default inputs, so the result above matches the widget exactly.
Sum to infinity for first term 10
S = 10 / (1 - r).
| Ratio (r) | Sum |
|---|---|
| 0.10 | 11.11 |
| 0.25 | 13.33 |
| 0.50 | 20.00 |
| 0.75 | 40.00 |
| 0.90 | 100.00 |
Mathematical functions reference: US National Institute of Standards and Technology (NIST).
Infinite Geometric Sum Calculator: frequently asked questions
When does an infinite geometric series have a sum?
Only when the common ratio's absolute value is less than one. In that case the terms shrink toward zero quickly enough that the endless sum approaches a finite limit. If the ratio is one or larger in magnitude, the terms do not shrink and the series diverges, so no finite sum exists.
What is the common ratio?
It is the fixed factor by which each term is multiplied to get the next. In the series 10, 5, 2.5, 1.25 and so on, each term is half the one before, so the common ratio is 0.5. You find it by dividing any term by the term immediately before it.
Can the ratio be negative?
Yes, as long as its absolute value is below one. A negative ratio produces an alternating series whose terms switch sign, but it still converges and the same formula applies. For example, a first term of 10 with a ratio of minus 0.5 sums to 10 divided by 1.5.
How does this relate to repeating decimals?
A repeating decimal is an infinite geometric series in disguise. For instance 0.3333 repeating equals 3 tenths plus 3 hundredths plus 3 thousandths, a series with first term 0.3 and ratio 0.1, which sums to 0.3 over 0.9, exactly one third.
What is the sum formula?
The sum to infinity equals the first term divided by one minus the ratio: S = a / (1 - r), valid when the ratio lies strictly between minus one and one. For a first term of 10 and a ratio of 0.5 the sum is 10 over 0.5, which is 20.
Official sources
- Mathematical functions and integer-sequence reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.