What is the formula for the sum of an infinite geometric series?

S = a / (1 − r), where a is the first term and r is the common ratio. This formula is only valid when |r| < 1; otherwise the series diverges.

When does an infinite geometric series converge?

When the absolute value of the common ratio |r| is strictly less than 1. For example, r = 0.5 converges; r = 1, r = −1, or r = 2 all diverge.

Can the first term (a) be negative?

Yes. The first term can be any real number, positive or negative. The convergence condition depends only on |r|, not on a.

What happens when r = 0?

When r = 0, every term after the first is zero, so the sum equals a. The formula S = a / (1 − 0) = a confirms this.

What is the difference between a finite and infinite geometric series?

A finite geometric series has a fixed number of terms and always has a sum. An infinite series sums infinitely many terms and converges only when |r| < 1.

Infinite Geometric Series Calculator

Enter a first term and common ratio to find the sum of an infinite geometric series, or check if it diverges.

Series Parameters

Formula: S = a / (1 − r), valid when |r| < 1

First Term (a)

Common Ratio (r)

Quick Examples

Enter values and click Calculate.

First 8 Terms

n	Term value	Partial sum

Summary

An infinite geometric series converges to a finite sum when the absolute value of the common ratio is less than 1. The formula S = a / (1 − r) gives the exact sum, where a is the first term and r is the common ratio. When |r| ≥ 1 the series diverges — it grows without bound or oscillates. This calculator evaluates both cases instantly and shows the first several terms so you can verify your inputs.

How it works

Enter the first term (a) — any non-zero number.
Enter the common ratio (r) — the fixed multiplier between consecutive terms.
Click Calculate to evaluate the series.
If |r| < 1, the calculator applies S = a / (1 − r) and displays the exact sum.
If |r| ≥ 1, the series diverges and the tool explains why.
The first 8 terms of the series are displayed so you can sanity-check your inputs.

Use cases

Solve calculus homework problems involving infinite series convergence.
Find the total distance traveled by a bouncing ball that loses a fixed fraction of height each bounce.
Calculate the present value of a perpetuity in finance (constant payment, constant discount rate).
Verify that a geometric series converges before summing it analytically.
Explore how quickly partial sums approach the infinite sum as more terms are added.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu