How do you calculate this sum $\sum_{n=1}^\infty nx^n$?

Question

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.

Answer 1

remember

$$ \sum x^n = \frac{1}{1-x } $$

for $|x| < 1 $

Answer 2

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} $