Chapter 10.1, Problem 41E

To determine

To find:The geometric series that express the repeating decimal $0.\overline{234}$ and then find the sum.

Answer to Problem 41E

The repeating decimal $0.\overline{234}$ can be expressed as $0.234{\displaystyle \sum _{n=0}^{\infty }{\left(0.001\right)}^n}$ and the sum is $\frac{26}{111}$ .

Explanation of Solution

Given:

The repeating decimal is $0.\overline{234}$ .

Concept used:

The mathematical expression of infinite geometric series in general is written as $a+ar+a{r}^2+a{r}^3+\cdots$ where the first term of the series is $a$ and the common ratio between two consecutive terms is $r$ .

The convergence test of series states that the geometric series converges if $\left|r\right|<1$ otherwise the series diverges where $r$ is the constant ratio.

Furthermore, the convergence test of geometric series also ensures that the convergent infinite geometric series ${\displaystyle \sum _{n=1}^{\infty }a}{r}^{n-1}$ converges to $\frac{a}{1-r}$ .

Calculation:

The given repeating decimal is $0.\overline{234}$ .

Rewrite the repeating decimal $0.\overline{234}$ as follows:

  $\begin{array}{c}0.\overline{234}=0.234+0.234\left(0.01\right)+0.234{\left(0.01\right)}^2+\cdots +0.234{\left(0.01\right)}^n+\cdots \\ =0.234\left(1+0.01+{\left(0.01\right)}^2+{\left(0.01\right)}^2+\cdots \right)\\ =0.234{\displaystyle \sum _{n=0}^{\infty }{\left(0.001\right)}^n}\end{array}$

The given function fits the formula for infinite geometric series ${\displaystyle \sum _{n=1}^{\infty }a}{r}^{n-1}$ that converges to $\frac{a}{1-r}$ .

Here, the initial term is 1 and each sequential term is multiplied by $0.001$ and the last term is the $n^{th}$ term that is equal to $2^{n-1}{x}^n$ .

The values are obtained as $a=1,r=0.001$ .

Find the sum of the geometric series as follows.

  $\begin{array}{c}S=0.234\left(\frac{1}{1-0.001}\right)\\ =\frac{0.234}{0.999}\\ =\frac{234}{999}\\ =\frac{26}{111}\end{array}$

Therefore, the sum of the geometric series is $\frac{26}{111}$ .

Conclusion:

Thus, the repeating decimal $0.\overline{234}$ can be expressed as $0.234{\displaystyle \sum _{n=0}^{\infty }{\left(0.001\right)}^n}$ and the sum is $\frac{26}{111}$ .