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 infinite geometric series is 1.5

Step 1: Identify the first term (a) and common ratio (r)

The first term \(a\) is the first number in the series which is 1, and the common ratio \(r\) is the factor that each term is multiplied by to get the next term, which in this case is 1/3.

Step 2: Check if the absolute value of r is less than 1

Confirm this since the sum formula is valid only when the absolute value of the common ratio \(r\) is less than 1. Here, \(r = 1/3\) which is less than 1.

Step 3: Use the formula for the sum of an infinite geometric series

Apply the formula \(S = \frac{a}{1 - r}\). Substituting the values of \(a\) and \(r\) into the formula we get \(S = \frac{1}{1 - 1/3}\), which simplifies to \(S = 1.5\)