The sum of an infinite geometric progression (GP) is a finite value only if the absolute value of the common ratio (r) is less than 1 (|r| < 1). If this condition is met, the sum (S∞) can be 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 the limit of the sum of the first n terms of a GP as n approaches infinity. The formula for the sum of the first n terms 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
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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 to: S∞ = 1 / (1 - 1/2) = 2.
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Example 2: For the GP: 9, 6, 4, 8/3,... a = 9 and r = 2/3. Since |r| = 2/3 < 1, the sum is: S∞ = 9 / (1 - 2/3) = 27.
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Example 3 (Divergent Series): The GP: 2, 4, 8, 16,... has a = 2 and r = 2. Because |r| = 2 ≥ 1, the sum does not converge to a finite value; it diverges to infinity.
Important Considerations
It's crucial to remember that the formula S∞ = a / (1 - r) is only valid when |r| < 1. If |r| ≥ 1, the infinite geometric series diverges, meaning its sum is not a finite number.
