The formula for finding the sum of a convergent infinite geometric series is S = a / (1 - r), where 'a' represents the first term of the series and 'r' represents the common ratio.
Understanding the Formula
This formula works because, in a convergent geometric series, the absolute value of the common ratio 'r' must be less than 1 (i.e., -1 < r < 1). This condition ensures that as the series continues infinitely, the terms become progressively smaller and approach zero. Consequently, the sum of the infinite series converges to a finite value.
Key Components:
- a (First Term): The initial value in the geometric series.
- r (Common Ratio): The constant value by which each term is multiplied to obtain the next term. It's calculated by dividing any term by its preceding term.
- S (Sum): The value that the infinite geometric series converges to.
When the Formula Applies:
The formula S = a / (1 - r) is only valid when the geometric series is convergent, meaning |r| < 1. If |r| ≥ 1, the series either diverges (its sum goes to infinity) or oscillates, and this formula cannot be used to find a finite sum.
Example:
Consider the infinite geometric series: 1 + 1/2 + 1/4 + 1/8 + ...
Here, a = 1 (the first term) and r = 1/2 (the common ratio). Since |1/2| < 1, the series converges.
Using the formula:
S = a / (1 - r) = 1 / (1 - 1/2) = 1 / (1/2) = 2
Therefore, the sum of this infinite geometric series is 2.
In Summary
The formula S = a / (1 - r) is a powerful tool for determining the sum of a convergent infinite geometric series. Remember that it's crucial to verify that |r| < 1 before applying the formula to ensure that the series converges.
