Find the formula of a sequence using generating functions The given sequence is $\sum_{k=1}^{n}kx^{k}$.
How can I do this? I think I should start with the already known generating function:
$A(x) = \sum a^{n}x^{n} = \frac{1}{1 - ax}$
And then I should do something with this function so that I finally come up with the given one. But well, how can I do this?
$\sum_{k=1}^{n}kx^{k} = \sum_{k>=1} [A(x) - 1]k$
I could only get something like this, but first of all it's not the formula without sums, and second I'm not sure it's correct.
Could you help me?

An idea without (properly) generating functions:
$$\forall\,|x|<1\;,\;\;\frac1{1-x}=\sum_{k=0}^\infty x^k\stackrel{\text{derivative}}\implies\frac1{(1-x)^2}=\sum_{k=1}^\infty kx^{k-1}\;\;\ldots$$
