What is the formula for the sum of an infinite geometric series?

The formula for the sum of an infinite geometric series applies when the series meets certain criteria, specifically that the absolute value of the common ratio, denoted as ( r ), is less than 1. This is essential because when ( |r| < 1 ), the terms of the series decrease towards zero as you progress, allowing for a finite sum.

In this formula, ( S ) represents the sum of the infinite series, ( a ) is the first term of the series, and ( r ) is the common ratio between successive terms. The correct formula is expressed as ( S = \frac{a}{1 - r} ). This derives from summing the series:

For the first term ( a ), the second term is ( ar ), the third term is ( ar^2 ), and so on, leading to the series:

[ S = a + ar + ar^2 + ar^3 + \ldots ]

By factoring ( a ) from the series, you can express it as:

[ S = a(1 + r + r^2 + r^3 + \ldots) ]

The series in parentheses is a common geometric series that sums to ( \