Chapter 6.2, Problem 101E

In the following exercises, given that $\frac{1}{1-x}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{x}^n}$ use term-by-term differentiation or integration to find power series for each function centered at the given point.

101. $f(x)={\displaystyle {\int }_0^{x}1\text{n}tdt}$ where $1\text{n(}x\text{) = }{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{(-1)}^{n-1}}\frac{{(x-1)}^n}{n}$