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