Proof of the generating function of $1 -2x +3x^2 -4x^3+5x^4-6x^5+\cdots$ The ordinary generating function for the sequence $\{a_n\}_{n\geq0}$ where $a_n = (-1)^n\,n$ is
$$1 -2x +3x^2 -4x^3+5x^4-6x^5+\cdots = \frac{1}{(x+1)^2}$$
I can see from the geometric formula, and even from long division that
$$1 -x +x^2 -x^3+x^4-x^5+\cdots = \frac{1}{1+x}$$
but I'm not seeing the coefficients to explain $\frac{1}{(x+1)^2}.$

$$S_n = 1 -2x +3x^2 -4x^3+5x^4-6x^5+\cdots$$ $$\implies
  xS_n = x - 2x^2 + 3x^3 -\cdots$$
$$(1+x)S_n=1-x+x^2-x^3\cdots=\frac{1}{1+x}$$
