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Math Help - Extended Binomial Fractional Powers

  1. #1
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    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 (x+y)^{1/2} using the binomial theorem?
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  2. #2
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    Re: Extended Binomial Fractional Powers

    I suggest you read about the Binomial SERIES.

    Binomial series - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Extended Binomial Fractional Powers

    the way you're going to find:

    \begin{pmatrix}1/2\\k\end{pmatrix}

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

    \begin{pmatrix}1/2\\0\end{pmatrix}=1, by definition (we have no terms).

    \begin{pmatrix}1/2\\1\end{pmatrix}=\frac{1/2}{1!} = 1/2.

    \begin{pmatrix}1/2\\2\end{pmatrix}=\frac{(1/2)(1/2 - 1)}{2!} = -1/8.

    \begin{pmatrix}1/2\\3\end{pmatrix}=\frac{(1/2)(1/2 - 1)(1/2 - 2)}{3!} = 1/16.

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

    \begin{pmatrix}1/2\\k\end{pmatrix}=\frac{(1/2)(1/2 - 1)(1/2 - 2)\dots(1/2-k+1)}{k!}

    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).
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