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Chapter 8: Problem 41

Find the sum of each infinite geometric series. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$$

Short Answer

Expert verified
The sum of the infinite geometric series is \( \frac{2}{3} \).

Step by step solution

01

Identify the first term and the common ratio

The first term \(a\) is 1 and the common ratio \(r\) is \(-\frac{1}{2}\). This is found by dividing the second term by the first term or the third term by the second term.
02

Check if the series converges

The series converges because the absolute value of the common ratio \(-\frac{1}{2}\) is less than 1. Therefore, we can find the sum using the formula for the sum of an infinite geometric series.
03

Use the formula to find the sum

Plug the values of \(a\) and \(r\) into the formula \(S = \frac{a}{1-r}\). After substituting the given values, we get \(S = \frac{1}{1-(-\frac{1}{2})} = \frac{1}{1.5} = \frac{2}{3}\).

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