Extended Binomial Fractional Powers

I'm trying to understand how the binomial theorem can be applied to expand polynomials to fractional powers. I can sought of piece it together without using the theorem, but when I try to use the theorem I run into problems with fractional factorials.

For example how would I expand using the binomial theorem?

I suggest you read about the Binomial SERIES.

Binomial series - Wikipedia, the free encyclopedia

the way you're going to find:



is to subtract 0, then 1, then...up to k-1 from 1/2, multiply these all together, and then divide by k! so

, by definition (we have no terms).

.

.

.

after awhile, using the definition directly gets computationally intense, but:



usually, only the first few terms in the infinite series you get are actually written out (unless your exponent is a non-negative integer, in which case you actually get a finite series).