I have been trying to understand why the binomial theorem can work for negative and fractional indices.
I understand that when raising binomials to positive integral indices, each coefficient is simply the number of ways that you can pick each term (e.g. for $(x+y)^5$, if you want to make up $xy^4$ there are 5 brackets from which to pick the $x$, so this term will come up 5 times in the full expansion).
I am not sure if a way to understand the infinite expansion for negative and fractional indices exists, but if it does I would very much like to know! Otherwise, I haven't been able to find a proof that shows that the result of the expansion for positive integer powers is valid for negative or fractional indices. I would be very grateful if someone could point me in the direction of such a proof.