Binomial Theorem (idea) by Suvrat

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 ⁿC_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+x²+x³ + .....

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

binomial coefficient	proof of the ham sandwich theorem	one-word Pythagorean Theorem proof	Proof that A*0 = 0
Combinatorics	Pedant	Taylor Series	Pascal's Triangle
hamming distance	proof of Darboux's theorem	Goodwin's theorem	Sensei
Euler's identity	binostat	binomial distribution	Dedekind cut
idiot-proof containers	Euclid's algorithm	Proof of the Myhill Theorem	Binominal
Binomial	God Lives Underwater	factorial