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Thread: proving question?

  1. #1
    Member
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    proving question?

    Let $\displaystyle f(t)= e^{t^2/2}$
    by taylor series: $\displaystyle f(t)= \sum\limits_{j = 0}^\infty \frac{(t^2/2)^j}{j!}$= $\displaystyle \sum\limits_{j = 0}^\infty \frac{t^{2j}}{2^jj!}$
    So $\displaystyle f'(0)= 0$
    $\displaystyle f''(0)= 1$
    $\displaystyle f'''(0)=0$... where $\displaystyle f^{(n)}(0)=0$ when n is odd and $\displaystyle f^{(n)}(0)=1$ when n is even.
    Differentiating $\displaystyle \frac{t^{2j}}{2^jj!}$ w.r.t (t) = $\displaystyle \frac{2jt^{2j-1}}{2^jj!}$
    when n=2,t =1, give $\displaystyle \frac{2j}{2^jj!}$
    when n=4, t=1, give $\displaystyle \frac{(2j)(2j-1)}{2^jj!}$
    and so on
    This is quite obvious but i have difficulty putting in word to show that
    $\displaystyle f^{(n)}(0)= 0$ when n is odd and
    $\displaystyle f^{(n)}(0)= \frac{(2j)!}{2^jj!}$ when n=2j
    Any better way to show?
    Last edited by noob mathematician; Apr 6th 2009 at 08:17 AM.
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  2. #2
    MHF Contributor
    Joined
    Mar 2007
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    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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