Chapter 9.1, Problem 70E

To determine

To find: Out of the given four options which of the function represents the given power series.

Answer to Problem 70E

The correct option is option $\left(D\right)$ .

Explanation of Solution

Given information: The function $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\left(x-1\right)}^n}$ is given.

Since the given series is in the form ${\displaystyle \sum _{n=0}^{\infty }a}{r}^n=a+ar+a{r}^2+\cdots +\cdots a{r}^n+\cdots$ , it is an infinite geometric progression, where the common ratio $r$ is $x-1$ and the first term $a$ is $1$ .

For the series to be convergent, $\left|r\right|=\left|x-1\right|<1$ .

Which gives ,

  $\left|x-1\right|<1$

Since, according to the absolute rule, if $\left|u\right|<a,a>0\text{then}-a<u<a$ , $\left|-\left(x+1\right)\right|<1$ takes the form,

  $\begin{array}{c}-1<x-1<1\\ x-1>-1\text{ and }x-1<1\\ x>0\text{ and }x<2\\ 0<x<2\end{array}$

For the obtained value of x, the series is convergent. Hence, determine the sum of the given series by substituting $1$ for $a$ and $x-1$ for $r$ into $S=\frac{a}{1-r}$ and denote it by $f\left(x\right)$ .

  $\begin{array}{c}f\left(x\right)=\frac{1}{1-\left(x-1\right)}\\ =\frac{1}{1-x+1}\\ =\frac{1}{2-x}\end{array}$

Therefore, the function that represents the given series is $f\left(x\right)=\frac{1}{2-x}$ .

As a result, the correct choice is $\left(D\right)$ .