Convergence of the series nx^n

[Felipe Lincoln]

Homework Statement

By finding a closed formula for the nth partial sum $s_n$,
show that the series $s=\sum\limits_{n=1}^{\infty}nx^n$ converges to $\dfrac{x}{(1-x)^2}$ when $|x|<1$ and diverges otherwise.

Homework Equations

Maybe $s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}$ when $|x|<1$

The Attempt at a Solution

Finding the $s_n$
$s_n + 1=1 + \sum\limits_{k=1}^{n}(\sqrt[k]{k}x)^k = 1 + x + 2x^2 + 3x^3 + \dots +nx^n= \dfrac{1-(\sqrt[n]{n}x)^{n+1}}{1-\sqrt[n]{n}x}$
but I don't know if I can get anywhere from here, tried several ways and had no success.

 

[FactChecker]

If you know the n'th partial sum of the power series,
$\sum_{i=1} ^{i=n-1} x^i = \frac {(1-x^n)}{(1-x)}$
, can you take the derivative of both sides and use that?

 

[Felipe Lincoln]

If you know the n'th partial sum of the power series,
$\sum_{i=1} ^{i=n-1} x^i = \frac {(1-x^n)}{(1-x)}$
, can you take the derivative of both sides and use that?

I would never think of that! I'll try

EDIT: I got it, thank you so much!