Finding the sum of $1 \cdot 2x + 2 \cdot 3x^{2} + 3 \cdot 4x^{3} + \dots$ . What is the sum of $1 \cdot 2x + 2 \cdot 3x^{2} + 3 \cdot 4x^{3} + \dots$ ?
I have been stuck on this problem with no direction. I have tried multiplying the sequence with $x$ and trying out $S-Sx$ but have gotten nowhere. Any help?
Thanks.

Hint:
$F'(x)=S=1\cdot2x+2\cdot3x^2+3\cdot4x^3+\cdots$
$F(x)=x^2+2x^3+3x^4+\cdots=x^2(1+2x+3x^2+\cdots)=\dfrac{x^2}{(1-x)^2}$
