Okay and for fractional powers there is another version of the binomial theorem. The first thing to notice is that the rth binomial coefficient can be written as
So if instead of nCk you use the above formula for the binomial coefficients you will end up with an infinite binomial series for any power.
Here's an example. Consider (1-x)-1. Its easy to show using the above expression for the binomial coefficients(with n = -1) that the series expansion here is
1+x+x2+x3 + .....
Of course this form of the theorem cannot be proved by induction. The Taylor's Theorem could be used though.