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Math Help - proving question?

  1. #1
    Member
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    Oct 2008
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    proving question?

    Let f(t)= e^{t^2/2}
    by taylor series: f(t)= \sum\limits_{j = 0}^\infty \frac{(t^2/2)^j}{j!}= \sum\limits_{j = 0}^\infty \frac{t^{2j}}{2^jj!}
    So f'(0)= 0
    f''(0)= 1
    f'''(0)=0... where f^{(n)}(0)=0 when n is odd and f^{(n)}(0)=1 when n is even.
    Differentiating \frac{t^{2j}}{2^jj!} w.r.t (t) = \frac{2jt^{2j-1}}{2^jj!}
    when n=2,t =1, give \frac{2j}{2^jj!}
    when n=4, t=1, give \frac{(2j)(2j-1)}{2^jj!}
    and so on
    This is quite obvious but i have difficulty putting in word to show that
    f^{(n)}(0)= 0 when n is odd and
    f^{(n)}(0)= \frac{(2j)!}{2^jj!} when n=2j
    Any better way to show?
    Last edited by noob mathematician; April 6th 2009 at 09:17 AM.
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  2. #2
    MHF Contributor
    Joined
    Mar 2007
    Posts
    1,240

    Question

    I see in your post a statement of a function and a statement of the Taylor polynomial, along with some conclusions about derivatives.

    But what are you supposed to be proving?
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