Chapter 11.6, Problem 10OE

To determine

To calculate:

Common ratio of given geometric series and sum series has.

Answer to Problem 10OE

Common ratio of given geometric series is $\frac{1}{10}$ and sum is $\frac{1}{9}$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{{10}^n}}$

Calculation:

In order to find out common ratio of given number,

  $\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{{10}^n}}\\ {\displaystyle \sum _{n=1}^{\infty }{\left(\frac{1}{10}\right)}^n}\end{array}$

Common ratio $\left(r\right)$ is,

  $r=\frac{1}{10}$

Converge if $-1<r<1$

Sum of given geometric series is calculated as,

  $=\frac{a^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}=\frac{a^n}{1-r}\\ =\frac{{\left(\frac{1}{10}\right)}^1}{1-\left(\frac{1}{10}\right)}\\ =\frac{0.1}{9/10}\\ =\frac{1}{9}\end{array}$

Therefore, Common ratio of given geometric series is $\frac{1}{10}$ and sum is $\frac{1}{9}$ .