motivation for $1/(1-x)$ used in generating functions A closed form for the infinite series $1+x+x^2+\ldots+x^n$ works out quickly like this:
$S_n = 1+x+x^2+\ldots+x^n$
$xS_n = x+x^2+x^3\ldots+x^{n-1}$
$S_n-xS_n=S_n(1-x)=1+x-x+x^2-x^2\ldots+x^n-x^n-x^{n-1} = 1-x^{n+1}$
$S_n=\frac{1-x^{n+1}}{1-x}$
When $|x|<1$ and $n$ is large, this becomes $\frac{1}{1-x}$.
I've been trying to udnerstand the basics of generating functions, and every account seems to take the fact that $\frac{1}{1-x}$ generates $\sum{x^n}$ for granted, saying that we don't worry about convergence and leaving it at that.
So why we don't use $\frac{1-x^{n+1}}{1-x}$ to generate $\sum{x^n}$ instead? I'm looking for a separate explanation or derivation explaning the general use of $\frac{1}{1-x}$.

A generating function is a formal power series. The set of formal power series forms a ring and in this ring $1-x$ has an inverse, which is $1+x+x^2+\cdots$.
