Chapter 10.1, Problem 71E

To determine

To choose: The correct option for the given series.

Answer to Problem 71E

The correct option for the given series is (D)

Explanation of Solution

Given information:

  $f(x)={\displaystyle \sum }_{n=0}^{\infty }{(x-1)}^n$

Calculation:

The given function is $f(x)={\displaystyle \sum }_{n=0}^{\infty }{(x-1)}^n$ it is a geometric progressive with initial terms is 1 and common ratio is equal to $x-1$ .

The sum of the infinite geometric series whose ratio is less than $1S=\frac{a}{1-r}$ where, $a$ is initial term and $r$ is the common ratio

Therefore, sum of the series is

  $\begin{array}{cc}f(x)& =\frac{1}{(1-(x-1))}\\ & =\frac{1}{2-x}\end{array}$

Replacing $t$ with $xf(t)=\frac{1}{2-t}{{\displaystyle \int }}_0^{x}f(t)dt={{\displaystyle \int }}_0^x\left(\frac{1}{2-t}\right)dt{{\displaystyle \int }}_0^{x}f(t)dt=-\ln (|x-2|)+\ln 2$

  ${{\displaystyle \int }}_0^{x}f(t)dt=-\ln \left(\frac{|x-2|}{2}\right)$

. The correct option for the given series is (D)