In Exercises 37–44, find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of the form a + ar + ar^2 + ar^3 + ... where 'a' is the first term and 'r' is the common ratio. This series continues indefinitely, and its convergence depends on the absolute value of 'r'. If |r| < 1, the series converges to a finite value.

Sum of an Infinite Geometric Series

The sum S of an infinite geometric series can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula is applicable only when the series converges, meaning |r| must be less than 1.

Convergence of Series

Convergence refers to the behavior of a series as the number of terms approaches infinity. For an infinite geometric series, convergence occurs when the common ratio 'r' satisfies the condition |r| < 1, ensuring that the terms decrease in magnitude and the series approaches a specific finite sum.