Convert Power Series to function

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I tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page.

Appreciate your help!

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2 Answers

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Hint: Why not look at $\sum \left(1-\frac{1}{n+1}\right)x^n$? Two series, one very familiar, the other almost as familiar.

Added: We have $\sum_1^\infty x^n=\frac{x}{1-x}$.

Also, $-\sum_1^\infty \frac{x^n}{n+1}=\frac{1}{x}\sum_1^\infty -\frac{x^{n+1}}{n+1}.$

Finally, $\sum_1^\infty -\frac{x^{n+1}}{n+1}=\ln(1-x)+x$.

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@Andre Nicolas

I got something totally differet. Separated as recommended for two series. and got: $\frac{x}{1-x} - \frac{1}{x(1-x)^2}$

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