The idea
When the magnitude of r is less than one, the terms shrink toward zero and the partial sums approach a finite limit. The infinite-sum formula follows from the finite-sum formula because rⁿ approaches zero. Terms approaching zero are necessary for convergence, but this alone does not establish convergence for every type of series.
Worked example
Find the sum of 12 + 6 + 3 + 1.5 + … .
- The first term is 12 and the common ratio is ½.
- Since |½| < 1, the geometric series converges.
- S∞ = 12/(1 − ½) = 24.
A mistake to avoid
Do not apply the infinite-sum formula when |r| ≥ 1. A result from substituting into a formula is not evidence that the sum exists.
Check your understanding
Does 8 − 4 + 2 − 1 + … converge, and what is its sum?
Show the explanation
Yes. r = −½, so |r| < 1. Its sum is 8/(1 + ½) = 16/3.
Explore the lesson
This guide connects to Sum of an infinite convergent geometric series in the IB Mathematics: Analysis and Approaches course. The first two published lessons per subject are free previews; further interactive lessons require an account.