How do you calculate this sum $\sum_{n=1}^\infty nx^n$? I can not find the function from which I have to start to calculate this power series.
 $$\sum_{n=1}^\infty nx^n$$
Any tips?.
Thanks.

We can do it another way.
$S = x + 2x^2 + 3x^3 + \ldots $
It can be written as 
$ \Rightarrow S = (x + x^2 + x^3 + \ldots)+(x^2 + x^3 + \ldots)+(x^3 + \ldots)+\ldots $
$\Rightarrow S = (x + x^2 + x^3 + ...)+x(x + x^2 + ...)+x^2(x + ...) + \ldots $
$\Rightarrow S = ( 1+x+x^2+ .. )\times( x+x^2+.. )$
$\Rightarrow S = \frac{1}{1-x}\times\frac{x}{1-x}$
$\Rightarrow S = \frac{x}{(1-x)^2} $
