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Homework Help: Binomial expansion comparison with legendre polynomial expansion

  1. Apr 15, 2012 #1

    I've been working on this question which asks to show that

    [itex]{{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}}[/itex]

    So first taking the n derivatives of the binomial expansions of (x2-1)n



    and comparing it with



    I'm having trouble with the final part,

    It's clear that there's a factor of 1/n!2n difference between them but also

    the Pn(x) series has m=0...n/2, and also xn , where as the n'th derivative series has k=0...n and x2n.

    How can you rewrite one in terms of the other so they both have the same sum limits?

    I've tried setting k=2s in the n'th derivative series and a bunch of other similar changes, but non will change the n'th powers of x.

    The reason I noticed this was because the last terms of the series arn't the same,

    the first series has last term, (-n)! on the bottom, means 1/infinity right?


    and the second


    Have I made a mistake early on or is there a clever way to combine the two series?


  2. jcsd
  3. Apr 16, 2012 #2

    I like Serena

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    Homework Helper

    Hi linda300! :smile:

    The factor 1/n!2n is already present in your first equation, so that is not a difference.

    When you took the n-th derivative, you didn't lower the power of x by n, so you should have xn-2k instead of x2n-2k

    Finally, when you take the derivative of x0 you should get zero, and not a negative power of x.
    So you should leave out the first n/2 terms, since they are all zero.
  4. Apr 16, 2012 #3
    Thanks heaps for taking the time to find my silly mistakes!

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