What is a geometric series?

A geometric series is the sum of a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example: 2 + 6 + 18 + 54 + ... with first term 2 and ratio 3.

When does an infinite geometric series converge?

An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1, i.e., |r| < 1. The sum equals a / (1 - r). If |r| ≥ 1, the series diverges (grows without bound).

What formula is used for a finite geometric series?

The sum of n terms is S = a × (1 - r^n) / (1 - r) when r ≠ 1. When r = 1, every term equals a and the sum is simply S = n × a.

Can the common ratio be negative?

Yes. A negative ratio causes the series to alternate between positive and negative values. Convergence still requires |r| < 1, so a ratio like −0.5 converges while −2 diverges.

What happens when r = 0?

When r = 0 the second and all subsequent terms are zero, so the sum of any finite series is just the first term a. An infinite series with r = 0 also equals a.

Geometric Series Sum

Calculate the sum of a finite or infinite geometric series from the first term and common ratio.

Series Parameters

First term (a)

Common ratio (r)

For infinite series, |r| must be < 1 to converge.

Number of terms (n)

Series Sum

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First term (a)

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Common ratio (r)

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First terms of the series

Press Calculate to see terms.

Summary

A geometric series is the sum of terms in a geometric sequence where each term is found by multiplying the previous one by a fixed ratio. This calculator handles both finite series (summing a specific number of terms) and infinite series (converging sums when the absolute value of the ratio is less than 1).

How it works

Enter the first term (a) of the series.
Enter the common ratio (r) — each term equals the previous term multiplied by r.
For a finite series, enter the number of terms (n) and click Calculate.
Toggle to Infinite mode to compute the converging sum when |r| < 1.
The tool displays the sum, the formula used, and the first several terms of the series.

Use cases

Calculate compound interest or exponential growth sums over time.
Verify textbook problems involving geometric progressions.
Determine the total distance traveled in a bouncing ball physics problem.
Model population or investment growth with a constant growth factor.
Compute the present value of a perpetuity using the infinite series formula.
Check convergence of a geometric series for calculus coursework.

Frequently Asked Questions

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