Problem 42 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sum of the series is 9/4.

Step 1: Identify the first term and the common ratio

In a geometric series, the first term 'a' is the first number in the series and the common ratio 'r' is the ratio of any term to its preceding term. In this series, the first term a = 3 and the common ratio r = -1/3 as each term alternates in sign and is a third of the previous term.

Step 2: Apply the formula for the sum of an infinite geometric series

The sum S of an infinite geometric series is given by the formula \( S = \frac{a}{1-r} \) where a is the first term and r is the common ratio.

Step 3: Compute the sum

By substituting a = 3 and r = -1/3 into the formula, it becomes \( S = \frac{3}{1 -(-1/3)} \). After simplifying the denominator it becomes \( S = \frac{3}{4/3} \), which simplifies to: \( S =9/4 \).