The sum of an infinite geometric progression (GP) is a finite number only if the absolute value of the common ratio (r) is less than 1 (|r| < 1). If this condition is met, the sum (S∞) is calculated using the formula:
S∞ = a / (1 - r)
Where:
- a is the first term of the GP.
- r is the common ratio of the GP.
Understanding the Formula
This formula arises from considering the limit of the sum of the first n terms of the GP as n approaches infinity. The formula for the sum of the first n terms of a GP is:
Sn = a(1 - rn) / (1 - r)
When |r| < 1, rn approaches 0 as n approaches infinity. Therefore, the term a(1 - rn) / (1 - r) simplifies to a / (1 - r).
Examples
-
Example 1: Consider the GP: 1, 1/2, 1/4, 1/8, ...
- Here, a = 1 and r = 1/2. Since |r| = 1/2 < 1, the sum converges.
- S∞ = 1 / (1 - 1/2) = 2
-
Example 2: Consider the GP: 1, 2, 4, 8, ...
- Here, a = 1 and r = 2. Since |r| = 2 > 1, the sum diverges (it goes to infinity). The formula for an infinite sum is not applicable in this case.
When the Sum Diverges
If |r| ≥ 1, the terms of the GP do not approach zero, and the sum of the infinite GP does not converge to a finite value. Instead, the sum either approaches positive or negative infinity, or it oscillates. The formula above is not valid for such cases.
The provided references confirm the formula and its limitations, emphasizing the requirement that |r| < 1 for a finite sum to exist. Several references highlight the importance of this condition for the sum to converge.
