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Thread: Binomial theorem for negative/rational exponents

  1. #1
    Member Greengoblin's Avatar
    Feb 2008

    Binomial theorem for negative/rational exponents

    We have the binomial theorem:

    $\displaystyle (x+y)^n = \sum_{r=0}^n\ ^nC_r x^n y^{n-r}$

    which makes perfect sense to me for nonnegative integers.

    Firstly, I don't understand how to extend this to the case of negative integer exponents, since I thought it would simply be:

    (x+y)^{-n}=((x+y)^n)^{-1}=\frac{1}{(x+y)^n}=\frac{1}{\sum_{r=0}^n\ ^nC_rx^ny^{n-r}}$

    This is fine, since the denominator still makes sense using $\displaystyle ^nC_r$, and we just have 1 over a polynomial, which also makes sense except for when p(x)=0.

    However, what I've read talks about an infinite series:

    $\displaystyle (x+y)^n=\sum_{r=0}^{\infty}\ ^nC_rx^ry^{n-r}$

    But I don't understand why this is needed if we can just have 1 over a polynomial.

    I aso don't get the case for rational exponents, but want to understand this first.
    Last edited by Greengoblin; Dec 2nd 2008 at 02:42 AM.
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