Chapter 10.1, Problem 37E

To determine

Find the value of $x$ .

Answer to Problem 37E

The value of $x$ is $\frac{19}{20}$ .

Explanation of Solution

Given Information:

The series is ${\displaystyle \sum _{n=0}^{\infty }{x}^n}=20$ .

Formula Used:

The sum of the geometric series when ration is less than 1:

  $S=\frac{a}{1-r}$ , where $a$ is initial term and $r$ is the common ratio.

Calculation:

The series is ${\displaystyle \sum _{n=0}^{\infty }{x}^n}=20$ .

Rewrite the series as:

  ${\displaystyle \sum _{n=0}^{\infty }{x}^n}=1+x+x^2+x^3+x^4+\cdots$

In the series ${\displaystyle \sum _{n=0}^{\infty }{x}^n}$ , $a=1$ and $r=x$ .

Find the value of $x$ .

Substitute 1 for $a$ and $x$ for $r$ in the formula $\frac{a}{1-r}$ .

  $\begin{array}{c}S=\frac{1}{1-x}\\ 20=\frac{1}{1-x}\\ 20-20x=1\end{array}$

Simplify the above equation.

  $\begin{array}{c}20-20x=1\\ 20x=19\\ x=\frac{19}{20}\end{array}$

Hence, the value of $x$ is $\frac{19}{20}$ .