<s>[INST] 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.
 [/INST] Expanding my hint, you have:
$$\frac{s}{x} = \frac{\text{d}^2}{\text{d} x^2}\left[ x^2 + x^3 + x^4 + x^5 + \cdots \right] = \frac{\text{d}^2}{\text{d} x^2}\left[ 1 + x + x^2 + x^3 + x^4 + x^5 + \cdots \right] = \frac{\text{d}^2}{\text{d} x^2}\left[ \frac{1}{1 - x} \right] = \frac{2}{(1 - x)^3}$$
(you can add $1+x$ to the sum in square brackets because $1+x$ has second derivative equal to $0$), hence:
$$ s = \frac{2x}{(1 - x)^3}\;.$$
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