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Chapter 9: Problem 57

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

Short Answer

Expert verified
A geometric series is a series 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 if its common ratio \( r \) is strictly between -1 and 1 (\( -1 < r < 1 \)). The formula for the sum of a convergent geometric series is \( S = a / (1 - r) \), where \( a \) is the first term in the series, and \( r \) is the common ratio.

Step by step solution

01

Definition of Geometric Series

A geometric series is a series 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.
02

Convergence of Geometric Series

A geometric series converges if its common ratio is strictly between -1 and 1, that is, \( -1 < r < 1 \), where \( r \) is the common ratio.
03

Formula for the Sum of the Convergent Geometric Series

The formula for the sum of a convergent geometric series is given by \( S = a / (1 - r) \), where \( a \) is the first term in the series, and \( r \) is the common ratio.

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