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

• A. S = a + (1 - r)
• B. S = a / (1 + r)
• C. S = a / (1 - r), where a is the first term and r is the common ratio
• D. S = a / (1 + 2r)

Answer: (C)

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 ( \