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?

Method 1.  Differentiate
$$1-x+x^2-x^3+\cdots=\frac1{1+x}\ .$$
Method 2.  Write the sum out this way:
$$\eqalign{
  1-x+x^2-x^3+x^4-\cdots&\cr
  {}-x+x^2-x^3+x^4-\cdots&\cr
  {}+x^2-x^3+x^4-\cdots&\cr
  {}-x^3+x^4-\cdots&\ .\cr}$$
Adding up the first row, then the second and so on by geometric series gives
$$\frac1{1+x}-\frac{x}{1+x}+\frac{x^2}{1+x}-\frac{x^3}{1+x}+\cdots$$
which is another geometric series.
