Chapter 8.4, Problem 30E

To determine

Check whether the given result is true or false.

Answer to Problem 30E

Thegiven result is true.

Explanation of Solution

Given:

The given expression is $\mathrm{0.999...}=1$

Calculation:

Write the repeating decimal as an infinite geometric series.

  $0.9+0.09+0.009+\cdot \cdot \cdot$

The sum of an infinite geometric series with first term $a_1$ and common ratio $r$ is

  $S=\frac{a_1}{1-r}$ , where $r<1$ .

If $r\ge 1$ , then the series has no sum.

In the given series $a_1=0.9$ and $r=\frac{0.09}{0.9}=0.1<1$ .

  $\begin{array}{l}S=\frac{0.9}{1-0.1}\\ =\frac{0.9}{0.9}\\ =1\end{array}$

Hence the given result is true.