Geometric Series Sum Calculator
A geometric series sums a sequence of numbers in which each term is multiplied by a fixed constant ratio r. The finite sum of n terms starting from first term a is S = a * (1 - r^n) / (1 - r), provided r is not equal to 1. When |r| is less than 1, the terms shrink toward zero and the infinite sum converges to a / (1 - r). Geometric series underpin compound interest calculations, annuity pricing, signal processing (the z-transform), and the analysis of repeating decimals. Enter the first term, common ratio, and number of terms to get the finite sum and the infinite series limit when it exists.
Geometric series formulas
Finite: S = a * (1 - r^n) / (1 - r), r not equal to 1
When r = 1: S = n * a
Infinite (|r| < 1): S = a / (1 - r)
The finite formula is derived by writing out S and r*S, then subtracting to cancel all intermediate terms. The infinite formula follows by taking the limit as n approaches infinity when |r| is less than 1, because r^n approaches zero.
Geometric series properties
- When |r| < 1, the terms decrease in magnitude and the infinite series converges.
- When |r| > 1, the terms grow without bound and the series diverges.
- When r = -1, the terms alternate between +a and -a; the series does not converge.
- The repeating decimal 0.333... = 3/10 + 3/100 + ... = (3/10) / (1 - 1/10) = 1/3.
- The present value of a perpetuity uses the infinite geometric series formula.
Geometric series: frequently asked questions
What is a geometric series?
A geometric series is the sum of the terms of a geometric sequence, where each term is multiplied by a constant ratio r. For example, 2 + 6 + 18 + 54 is a geometric series with first term a = 2 and common ratio r = 3.
What is the formula for the finite geometric series sum?
S = a * (1 - r^n) / (1 - r) for r not equal to 1. When r = 1, all terms are equal to a so S = n * a. Where a is the first term, r is the common ratio, and n is the number of terms.
When does the infinite geometric series converge?
The infinite sum converges to S = a / (1 - r) when |r| < 1. For |r| >= 1, the terms do not shrink to zero and the series diverges (grows without bound in magnitude).
What happens when r is negative?
When r is negative, the terms alternate in sign. The sum formula still applies: S = a(1-r^n)/(1-r). For example, 1 - 1/2 + 1/4 - 1/8 with a = 1, r = -0.5, n = 4 gives S = 1*(1-(-0.5)^4)/(1-(-0.5)) = (1-0.0625)/1.5 = 0.625.
How is a geometric series used in finance?
The present value of an annuity is a geometric series where each future payment is discounted by (1+i)^t. The formula PV = PMT * (1 - (1+i)^(-n)) / i is derived directly from the finite geometric series formula.
Official sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.