The sequence $\displaystyle b_n=1+2x+3x^2+...+nx^{n-1}, |x|<1$ converges to $\displaystyle (1-x)^{-2}$

I know how to prove that it converges to that value, but I'm unsure of how to prove that the sequence converges, ie, that the limit exists. I thought maybe trying to prove that the sequence is contractive would work, but I couldn't get a proof using that method. Does anyone know of another way to show that it converges?