# Showing/finding sumfunction

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?

• "For $x \in \Bbb{R}$" your power series does not converge for $x > 1$. – Clement Yung May 28 at 22:01
• Are you asking how to derive the closed form for the series, or are you asking how to determine for which real numbers it converges? – Brian M. Scott May 28 at 22:08

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}