Infinite Geometric Series Calculator

Q: What happens when |r| ≥ 1?

The terms do not shrink, so the partial sums grow without limit or oscillate; the series diverges and has no sum.

Work out infinite geometric series instantly with clear inputs, formula shown and shareable results.

About Infinite Geometric Series Calculator

When |r| < 1 the terms shrink fast enough for the series to converge, and the sum is a/(1 − r). The partial sum a(1 − rⁿ)/(1 − r) approaches that limit geometrically.

Formulas

Sum to infinity

S = a / (1 − r), valid for |r| < 1

Partial sum

Sₙ = a(1 − rⁿ) / (1 − r)

Worked example

With the default values below, Infinite Geometric Series Calculator returns Sum to infinity of 16.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

First term a: 8
Common ratio r: 0.5
Also show the sum of this many termsaccepts 1–10,000: 10

Results

Sum to infinity: 16.00000000
Sum of the first n terms: 15.98437500
Tail still remaining: 0.0156250000
The nth term: 0.0156250000

Step by step

1Values used
2Sum to infinity
3Partial sum
4Sum to infinity= 16.00000000
5Sum of the first n terms= 15.98437500
6Tail still remaining= 0.0156250000
7The nth term= 0.0156250000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Sum of 8 + 4 + 2 + 1 + …?

8/(1 − 0.5) = 16.

What happens when |r| ≥ 1?