Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of the terms of a geometric sequence that continues indefinitely. It is defined by a first term (a) and a common ratio (r). The series converges to a finite value if the absolute value of the common ratio is less than one (|r| < 1). The formula for the sum of an infinite geometric series is S = a / (1 - r).

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. It is calculated by dividing any term by its preceding term. For the series given, identifying the common ratio is crucial to applying the formula for the sum of the series. If the common ratio is greater than or equal to one in absolute value, the series diverges and does not have a finite sum.

Convergence of Series

Convergence refers to the behavior of a series as the number of terms approaches infinity. An infinite series converges if the sum approaches a specific finite value. For geometric series, convergence is determined by the common ratio; if |r| < 1, the series converges. Understanding convergence is essential for determining whether the infinite series in the question has a finite sum.