Geometric Series Calculator

Q: What is a geometric series?

The sum of the terms of a geometric sequence.

Q: Finite sum formula?

Sₙ = a₁(1-rⁿ)/(1-r) when r ≠ 1; Sₙ = a₁n when r = 1.

Q: Infinite sum formula?

S∞ = a₁/(1-r) when |r| < 1. Diverges otherwise.

Q: When does it converge?

For |r| < 1, the infinite sum converges.

Q: What is a real example?

A ball bouncing to half height each time: the total distance converges.

Use our free geometric series calculator to find the finite sum Sₙ, the infinite sum S∞ (when |r| < 1), or the sum from term m to n.

Math

Mode

First Term (a₁)

Common Ratio (r)

Number of Terms (n)

From Term (m)

Formula

A geometric series is the sum of a geometric sequence. For |r| < 1, the infinite sum converges to S∞ = a₁/(1−r). The finite sum Sₙ = a₁(1−rⁿ)/(1−r) for r ≠ 1.

Sₙ = a₁(1−rⁿ)/(1−r), S∞ = a₁/(1−r) for |r| < 1

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Select a mode.
2Finite: enter a₁, r, n.
3Infinite: enter a₁, r (|r| must be < 1).
4Range: enter a₁, r, m, n.
5Click Calculate.

Examples

S₁₀: a₁=3, r=0.5

1.S₁₀ = 3(1-0.5¹⁰)/(1-0.5) = 3(0.999023)/0.5 = 5.994141

Result: 5.994141

S∞: a₁=3, r=0.5

1.S∞ = 3/(1-0.5) = 6
2.|0.5| < 1 → converges

Result: 6

Terms 3-6: a₁=3, r=0.5

1.S₆ = 5.90625, S₂ = 4.5
2.Sum terms 3-6 = 5.90625 − 4.5 = 1.40625

Result: 1.406250

Frequently Asked Questions

The sum of the terms of a geometric sequence.

Sₙ = a₁(1-rⁿ)/(1-r) when r ≠ 1; Sₙ = a₁n when r = 1.

S∞ = a₁/(1-r) when |r| < 1. Diverges otherwise.

For |r| < 1, the infinite sum converges.

A ball bouncing to half height each time: the total distance converges.

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CalculateMe Team

Last updated: 2026-07-14