The formula for the sum of an infinite geometric series is S∞ = a / (1 – r), where 'a' is the first term and 'r' is the common ratio.
An infinite geometric series is a geometric series with an infinite number of terms. However, the sum of such a series converges to a finite value only if the absolute value of the common ratio 'r' is less than 1 (i.e., |r| < 1). If |r| ≥ 1, the series diverges and does not have a finite sum.
Understanding the Components
- S∞: Represents the sum of the infinite geometric series.
- a: Represents the first term of the geometric series.
- r: Represents the common ratio, which is the factor by which each term is multiplied to get the next term. It is calculated by dividing any term by its preceding term.
Condition for Convergence
The formula is valid only when |r| < 1. This condition ensures that the terms of the series become progressively smaller, allowing the sum to approach a finite limit. If |r| ≥ 1, the terms either remain the same size or grow larger, leading to a diverging series.
Example
Let's consider the infinite geometric series: 1 + 1/2 + 1/4 + 1/8 + ...
- a (first term) = 1
- r (common ratio) = 1/2
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) calculates the sum of an infinite geometric series, but is only applicable when the absolute value of the common ratio 'r' is less than 1 (|r| < 1).
