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Math Help - Ominous Sum:

  1. #1
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    Ominous Sum:

    Question: Prove for some general function f(x) for which \forall a \epsilon \mathbb{R} : x\geqslant a \cap f(x) is uniformly continuous on [a,b],
     f(x)=\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))

    -i invented the problem via FTC following these steps
    f(x)=\frac{d}{dx}\int_{a}^{x}f(t)dt=\frac{d}{dx}(\ lim_{n->\infty}\Sigma_{k=1}^n(\frac{x-a}{n}f(\frac{kx+(n-k)a}{n}))) =\lim_{n->\infty}\Sigma_{k=1}^n\frac{d}{dx}((\frac{x-a}{n}f(\frac{kx+(n-k)a}{n}))
    =\lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx+( n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n}))))

    particularly letting a=x gives \lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx+(  n-k)x}{n}))))= \lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(x))=f(x)
    Last edited by JeffN12345; January 6th 2010 at 09:25 PM. Reason: Drexel28's reply
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JeffN12345 View Post
    Question: Prove \forall a \epsilon \mathbb{R} : x\geqslant a ,  f(x)=\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))
    WITHOUTusing The Fundamental Theorem Of Calculus

    -i invented the problem via FTC following these steps
    f(x)=\frac{d}{dx}\int_{a}^{x}f(x)dx=\frac{d}{dx}(\  lim_{n->\infty}\Sigma_{k=1}^n(\frac{x-a}{n}f(\frac{kx+(n-k)a}{n}))) =\lim_{n->\infty}\Sigma_{k=1}^n\frac{d}{dx}((\frac{x-a}{n}f(\frac{kx+(n-k)a}{n}))
    =\lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx+(  n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n}))))

    particularly letting a=x gives \lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx+(  n-k)x}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)x}{n}))))= \lim_{n->\infty}(\Sigma_{k=1}^n(\frac{1}{n}(f(x)+0*f'(x)))  )=f(x)
    You preformed some unjustified madness in there, Jeff.
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  3. #3
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    Is the question sensical now
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  4. #4
    Senior Member Shanks's Avatar
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    the proposition is false!
    further, your solution has some flaw, too.
    pay attention to the oder of derivative and limit!
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  5. #5
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    Quote Originally Posted by Shanks View Post
    the proposition is false!
    further, your solution has some flaw, too.
    pay attention to the oder of derivative and limit!
    I was told differentials and limits and sums are all linear operators so their order can be interchanged...
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  6. #6
    Senior Member Shanks's Avatar
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    No! This is not always the case.
    That is why I said "pay attention to the interchange of oder".
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