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Math Help - Coefficient of x^n

  1. #1
    MHF Contributor alexmahone's Avatar
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    Coefficient of x^n

    What is the coefficient of x^n in the power series form of \sqrt[3]{1-2x}?

    My attempt:

    \sqrt[3]{1-2x}=(1-2x)^\frac{1}{3}=\sum_{n\geq 0}\dbinom{\frac{1}{3}}{n}(-2x)^n

    \dbinom{\frac{1}{3}}{n}=\frac{\frac{1}{3}.\frac{-2}{3}.\frac{-5}{3}...\frac{-3n+4}{3}}{n!}

    How do I simplify this and find the coefficient of x^n?
    Last edited by alexmahone; August 20th 2011 at 07:07 PM.
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  2. #2
    Member Traveller's Avatar
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    Re: Coefficient of x^n

    How about differentiating first?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Coefficient of x^n

    Quote Originally Posted by alexmahone View Post
    What is the coefficient of x^n in the power series form of \sqrt[3]{1-2x}?
    \textrm{coef}\;(x^n)=(-2)^n\binom{1/3}{n}\quad (\forall n\geq 0) . For n\geq 2 we can simplify to obtain

    \textrm{coef}\;(x^n)=\ldots=-\frac{2^n}{3^n}\cdot \dfrac{2\cdot 5\cdot \ldots\cdot (3n-4)}{n!}
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