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

Name: Infinite Geometric Series Calculator
Availability: InStock
Author: Nham Vu

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

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

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: 2026-06-13 · Reviewed by Nham Vu