How to find closed form of a binomial series. When working on a problem, I needed to find the closed form of the infinite sequence:
$$1 - 2x + 3x^2 - 4x^3 + \cdots$$
I struggled with this for a while and eventually found, through the Internet, that it is equal to $(1+x)^{-2}$.
How could I have approached this in such a way that I would have arrived at the answer myself?

There are many ways to get the function from this series. One is to use negative binomial coefficients:
$$
\begin{align}
\sum_{k=0}^\infty(-1)^k\binom{k+1}{1}x^k
&=\sum_{k=0}^\infty(-1)^k\binom{k+1}{k}x^k\\
&=\sum_{k=0}^\infty\binom{-2}{k}x^k\\[4pt]
&=(1+x)^{-2}
\end{align}
$$

Another method would be to subtract the geometric series
$$
\frac1{1+x}=1-x+x^2-x^3+x^4-x^5+\dots
$$
to get
$$
\begin{align}
f(x)-\frac1{1+x}
&=-x+2x^2-3x^3+4x^4-\dots\\
&=-xf(x)
\end{align}
$$
Solving for $f(x)$ gives
$$
f(x)=\frac1{(1+x)^2}
$$
