Problem 41 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \(S = \frac{1}{1-(-\frac{1}{2})} = \frac{1}{1.5} = \frac{2}{3}\).