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Chapter 8: Problem 37

Find the sum of each infinite geometric series. $$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$$

Short Answer

Expert verified
The sum of the given infinite geometric series is 1.5.

Step by step solution

01

Identify the first term and the common ratio

The first term of the series \(a\) is 1 and the common ratio \(r\) is \( \frac{1}{3} \).
02

Apply the formula to find the sum

Substitute \(a=1\) and \(r=\frac{1}{3}\) into the formula \(S = \frac{a}{1-r}\). Hence, the sum \(S\) of the infinite geometric series is \(S = \frac{1}{1-\frac{1}{3}}\).
03

Simplifying the equation

Simplify the equation to find the sum. \(S = \frac{1}{\frac{2}{3}}\) which simplifies to \(S = 1.5\).

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