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Chapter 14: Problem 102

How do you determine if an infinite geometric series has a sum? Explain how to find the sum of such an infinite geometric series.

Short Answer

Expert verified
An infinite geometric series has a sum if the absolute value of the common ratio is less than 1. The sum of the series can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

Step by step solution

01

Identify the geometric series

A geometric series is a series where each term after the first one is found by multiplying the previous term by a fixed, non-zero number called the common ratio. We can write it in the form \(a, ar, ar^2, ar^3, ...\), where \(a\) is the first term, and \(r\) is the common ratio.
02

Check the common ratio

For the series to be summable, the common ratio must lie between -1 and 1 (exclusive), i.e., -1 < \(r\) < 1. If this condition is met, the geometric series is convergent (i.e., it has a finite sum). If not, the series is divergent and the sum cannot be found.
03

Calculate the sum

We use the formula to calculate the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \]where \(S\) is the total sum of the series, \(a\) is the first term and \(r\) is the common ratio.

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