Okay and for fractional powers there is another version of the binomial theorem. The first thing to notice is that the r^{th} binomial coefficient can be written as

(n(n-1)(n-2)(n-3)...(n-r+1))/r!

So if instead of ^{n}C_{k} 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+x^{2}+x^{3} + .....

Of course this form of the theorem cannot be proved by induction. The Taylor's Theorem could be used though.