Question

Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.

Short answer

A geometric series is defined as a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. It converges when the absolute value of the common ratio is less than 1. The sum of a convergent geometric series is given by \( S = \frac{a}{1 - r} \), where 'a' is the first term and 'r' is the common ratio of the series.

Step-by-step solution

Definition of Geometric Series

A geometric sequence or series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with common ratio 3.

Convergence of Geometric Series

A geometric series converges if the absolute value of the common ratio is less than 1 i.e. \( -1 < r < 1 \) where r is the common ratio.

Sum of Convergent Geometric Series

For a geometric series that converges, the sum of the series 'S' is given by: \( S = \frac{a}{1 - r} \), where 'a' is the first term and 'r' is the common ratio of the series.