We have the binomial theorem:

$\displaystyle (x+y)^n = \sum_{r=0}^n\ ^nC_r x^n y^{n-r}$

which makes perfect sense to me for nonnegative integers.

Firstly, I don't understand how to extend this to the case of negative integer exponents, since I thought it would simply be:

$\displaystyle

(x+y)^{-n}=((x+y)^n)^{-1}=\frac{1}{(x+y)^n}=\frac{1}{\sum_{r=0}^n\ ^nC_rx^ny^{n-r}}$

This is fine, since the denominator still makes sense using $\displaystyle ^nC_r$, and we just have 1 over a polynomial, which also makes sense except for when p(x)=0.

However, what I've read talks about an infinite series:

$\displaystyle (x+y)^n=\sum_{r=0}^{\infty}\ ^nC_rx^ry^{n-r}$

But I don't understand why this is needed if we can just have 1 over a polynomial.

I aso don't get the case for rational exponents, but want to understand this first.