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I want to evaluate the series $$\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}$$ I don't know how to do this by hand, but Mathematica tells me that the answer is: $$\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}=x$$

and this is indeed the answer I want. However, I still want to know how to do this by hand.

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HINT

We have that

$$\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}=\frac1{1+x}\sum_{n=0}^{\infty}n\left(\frac{x}{1+x}\right)^n$$

then recall that

$$\sum_{k=0}^\infty kr^{k}=\frac{r}{(1-r)^2}\;\;,\;\;|r|<1$$

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    $\begingroup$ You are too fast for the old man ! I was finishing typing the same when your answer came in. $\to +1$ $\endgroup$ – Claude Leibovici Jun 8 '18 at 10:19
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    $\begingroup$ @ClaudeLeibovici I'm sorry Claude, in that case you should show also your answer! Repetita iuvant :) $\endgroup$ – gimusi Jun 8 '18 at 10:20
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    $\begingroup$ Don't be sorry for anything ! You did it very well, that is it ! Cheers, my friend (if I may). $\endgroup$ – Claude Leibovici Jun 8 '18 at 10:33
  • $\begingroup$ Gimusi.Presque comme toujours.Formidable. $\endgroup$ – Peter Szilas Jun 8 '18 at 10:54
  • $\begingroup$ @ClaudeLeibovici I’m honored if I may be your friend! Thanks $\endgroup$ – gimusi Jun 8 '18 at 11:13

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