Infinite Geometric Series Calculator

Enter the first term a and common ratio r to find the sum of the infinite geometric series a + a r + a r² + …. It computes S = a / (1 − r) when the absolute value of r is below 1 and tells you whether the series converges or diverges.

Input

Enter the first term a and common ratio r of the infinite geometric series a + a r + a r² + …. It finds the sum S = a / (1 − r).

First term a

Common ratio r

The series converges and has a sum when the absolute value of the ratio is below 1. It diverges when that value is 1 or greater.

Result

Convergent series

Sum S

Common ratio r

0.5

Convergence

Converges

Partial sums of the first n terms

Terms n	Partial sum Sₙ
1	1
2	1.5
3	1.75
4	1.875
5	1.9375
6	1.96875
7	1.984375
8	1.9921875
9	1.99609375
10	1.99804688

As more terms are added the partial sums approach the total S.

The sum is S = a / (1 − r). The series converges only when the absolute value of the ratio is below 1 and diverges otherwise.

How it works

The sum of an infinite geometric series a + a r + a r² + … converges only when the absolute value of the common ratio r is below 1, and then equals S = a / (1 − r).
When the absolute value of r is 1 or greater the partial sums do not settle on a single value, so the series diverges. The exception is a first term of 0, where every term is 0 and the sum is 0.
The partial sums table lists the running total of the first n terms, S_n = a (1 − rⁿ) / (1 − r); for a convergent series these values approach the total S as n grows.

Reviews

Tell us what you think of this calculator.