Showing/finding sumfunction

Question

I will try again asking my question: I have $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. Then I have used wolframalpha finding the sum function: https://www.wolframalpha.com/input/?i=sum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7Bn*x%5E%7Bn%7D%7D%7Bn%2B1%7D And with Maple I have reduced the sum function to: $\frac{1}{1-x}+\frac{ln(1-x)}{x}$ for |x|<1. But how can I show it formally that this is the sum function?

Answer 1

Note that for $|x|<1$

$$\begin{align} \sum_{n=1}^\infty \frac{nx^n}{n+1}&=\sum_{n=1}^\infty x^n-\frac1x\sum_{n=1}^\infty \frac{x^{n+1}}{n+1}\\\\ &=\frac x{1-x}-\frac1x\int_0^x \sum_{n=1}^\infty t^n\,dt\\\\ &=\frac x{1-x}-\frac1x\int_0^x \frac{t}{1-t}\,dt \end{align}$$

Can you finish now?