Find sumfunction? Need hints

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I have $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. Then I have to find the sum function. I know there is many ways to find the sumfunction. But can I differentiate with respect to x and then find the sum of it? Then I have the differentiabel version of the sum function? Then I can take the integral from 0 to $\infty$?