Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is 1.5.

Step 1: Identify the first term and the common ratio

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

Step 2: 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 \(S = \frac{1}{1-\frac{1}{3}}\).

Step 3: Simplifying the equation

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