Geometric series sum to infinity: S∞ = a / (1 − r)
The geometric series sum to infinity is S∞ = a / (1 − r), for |r| < 1.
What each letter means
- \(S_\infty\) the sum to infinity
- \(a\) the first term
- \(r\) the common ratio, with |r| < 1
When to use it
When a geometric series goes on for ever and its terms shrink towards 0, for example a recurring decimal or the total distance travelled by a bouncing ball.
Worked example
A geometric series has first term 15 and sum to infinity 45. Find the common ratio.
- \(\dfrac{15}{1-r}=45\), so \(1-r=\dfrac13\)
Answer: \(r=\tfrac23\)
Common mistake
Using it when |r| ≥ 1. Then the terms do not shrink and the series has no sum to infinity.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | In the Edexcel formulae booklet |
| Edexcel International A Level | In Pearson’s IAL formulae booklet |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Sequences and series
- Revise IAL P2 sequences and series
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the geometric series sum to infinity formula?
The geometric series sum to infinity is S∞ = a / (1 − r), for |r| < 1. S_∞: the sum to infinity; a: the first term; r: the common ratio, with |r| < 1.
Is the sum to infinity of a geometric series given in the exam?
Edexcel A Level (9MA0): in the Edexcel formulae booklet. Edexcel International A Level: in Pearson’s IAL formulae booklet. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
When does a geometric series have a sum to infinity?
Only when −1 < r < 1. The terms then shrink towards 0 and the sums settle at u₁ / (1 − r).
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.