Chapter 10.1, Problem 3QC

Reasoning as in Example 7, what is the value of 0.3 + 0.03 + 0.003 + ….?

Example 7 Working with Series

Consider the infinite series

0.9 + 0.09 + 0.009 + 0.0009 + …,

where each term of the sum is $\frac{1}{10}$ of the previous term.

1. a. Find the sum of the first one, two, three, and four terms of the series.
2. b. What value would you assign to the infinite series 0.9 + 0.09 + 0.009 + …?

Solution

1. a. Let S_n denote the sum of the first n terms of the given series. Then

   S₁ = 0.9,

   S₂ = 0.9 + 0.09 = 0.99,

   S₃ = 0.9 + 0.09 + 0.009 = 0.999, and

   S₄ = 0.9 + 0.09 + 0.009 + 0.0009 = 0.9999.

2. b. The sums S₁, S₂ …, S_n form a sequence {S_n}, which is a sequence of partial sums. As more and more terms are included, the values of S_n approach 1. Therefore, a reasonable conjecture for the value of the series is 1: