The formula for the sum to infinity (S∞) of a geometric series is:
S∞ = a / (1 - r)
where:
- a is the first term of the series.
- r is the common ratio between consecutive terms.
This formula is valid only when the absolute value of the common ratio, |r|, is less than 1 (i.e., -1 < r < 1). If |r| ≥ 1, the series diverges, meaning it does not have a finite sum.
Several sources confirm this formula:
- Cuemath: Clearly states the formula S∞ = a/(1 – r) for an infinite geometric series.
- PW.Live: Reinforces the formula S∞ = a / (1 - r) for the sum of infinite terms in a geometric progression.
- Cuemath (Sum of Infinite GP): While also mentioning the formula for the sum of the first 'n' terms, it implicitly supports the infinity formula by its context.
- Mathsathome: Explicitly gives the formula S∞ = a₁/(1-r), using a₁ for the first term.
- Unacademy: Specifies S infinite = a/1-r as the formula for an infinite geometric progression.
Example:
Consider the geometric series: 1 + 1/2 + 1/4 + 1/8 + ...
Here, a = 1 and r = 1/2. Since |r| = 1/2 < 1, the sum to infinity exists and is:
S∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2
