<s>[INST] Summation $1-4x+9x^2-16x^3 + \cdots ?$ Taylor series gives $$\frac 1 {(1+x)^2}=1-2x+3x^2-4x^3+\cdots$$
is there a nice expression for $1-4x+9x^2-16x^3 + \cdots ?$
It would be helpful for a problem I am trying to solve.
 [/INST] For $|x| < 1$, we have
$$\begin{align}
\sum_{n=0}^\infty (n+1)^2 (-x)^n 
&= \sum_{n=0}^\infty \left(x\frac{d}{dx} + 1 \right)^2 (-x)^n
= \left(x\frac{d}{dx} + 1 \right)^2 \sum_{n=0}^\infty (-x)^n\\
&= \left(x\frac{d}{dx} + 1 \right)^2 \frac{1}{1+x}
 = \frac{d}{dx}\left[ x \frac{d}{dx}\left(\frac{x}{1+x}\right)\right]
 = \frac{d}{dx}\left[\frac{x}{(1+x)^2}\right]\\
&= \frac{1-x}{(1+x)^3}
\end{align}
$$
</s>