<s>[INST] Prove infinite series $$
\frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2}
$$
I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be $x>1$, the question doesn't specify. Putting the LHS into Wolfram Alpha doesn't generate the RHS (it times out).
 [/INST] Let's put this together cleanly:
For the sum to converge, $|x| > 1$ which means
 $$\sum_{n=0}^\infty x^{-n} = \frac{x}{x-1}$$
Squaring this gives
 $$\frac{x^2}{(x-1)^2} = \sum_{n=0}^\infty \sum_{m=0}^\infty x^{-(n+m)} $$
$$ = \sum_{r=0}^\infty (r+1)x^{-r}$$
(because each value of $(n+m)$ occurs $(n+m+1)$ times in the infinite sum (0 = 0+0, 1 = 1+0 or 0+1, 2=0+2 or 1+1 or 2+0, and so on).
Divide by $x$ gives:
$$ \frac{x}{(x-1)^2} = \sum_{r=0}^\infty (r+1)x^{-(r+1)} $$
$$ = \frac1{x} + \frac{2}{x^2} + \frac{3}{x^3} + ... $$
</s>