An infinite geometric progression (GP) series is a sum of an infinite number of terms where each term is obtained by multiplying the previous term by a constant value called the common ratio (r). The series only converges to a finite sum if the absolute value of the common ratio is less than 1 (|r| < 1).
Understanding the Formula
The sum of an infinite GP series is given by the formula:
S∞ = a / (1 - r)
where:
- S∞ represents the sum of the infinite series.
- a is the first term of the series.
- r is the common ratio.
This formula is valid only when |r| < 1. If |r| ≥ 1, the series diverges and the sum is infinite.
Examples
-
Convergent Series: Consider the series 1 + 1/2 + 1/4 + 1/8 + ... Here, a = 1 and r = 1/2. Since |r| = 1/2 < 1, the series converges. The sum is S∞ = 1 / (1 - 1/2) = 2.
-
Divergent Series: The series 1 + 2 + 4 + 8 + ... has a = 1 and r = 2. Since |r| = 2 > 1, the series diverges and does not have a finite sum.
Important Considerations
- As noted by multiple sources like Cuemath (https://www.cuemath.com/algebra/sum-of-an-infinite-gp/) and Byju's (https://byjus.com/maths/geometric-progression-sum-of-gp/), the convergence condition (|r| < 1) is crucial for the formula's validity. Without this condition, the sum is undefined.
- The sum of an infinite GP is not an approximation; it's the exact value the series approaches as the number of terms goes to infinity, provided it converges (https://www.quora.com/Isnt-the-sum-of-an-infinite-G-P-series-really-just-a-very-close-approximation-of-the-sum-of-that-series-as-we-keep-adding-more-and-more-terms).
