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Math Help - Weirstrass M-test (convergence) and derivative

  1. #1
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    Weirstrass M-test (convergence) and derivative

    Hi,

    I would like to know how to show the following:

    Let f(x)= summation of an (note that n is subscript) * (Sin nx)
    so f(x) = summation (an)(Sin nx).
    If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

    Can someone please help me on this? Thank you!
    I would also like to know what theorem on series of derivatives should we apply.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Do you go to Concordia university by any chance?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zxcv View Post
    Hi,

    I would like to know how to show the following:

    Let f(x)= summation of an (note that n is subscript) * (Sin nx)
    so f(x) = summation (an)(Sin nx).
    If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

    Can someone please help me on this? Thank you!
    I would also like to know what theorem on series of derivatives should we apply.
    What is the conditions for a series to be differentiable?
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  4. #4
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    Quote Originally Posted by zxcv View Post
    Hi,

    I would like to know how to show the following:

    Let f(x)= summation of an (note that n is subscript) * (Sin nx)
    so f(x) = summation (an)(Sin nx).
    If the series summation of n(an) is absolutely convergent, show that f is differentiable and f'(x)= summation of n(an)(cos nx).

    Can someone please help me on this? Thank you!
    I would also like to know what theorem on series of derivatives should we apply.

    Let  F_k(x) = \sum_{i=0}^k a_n sin(nx)

    Clearly  F'_k(x) = \sum_{i=0}^k na_ncos(nx)

    And there's a theorem that says if  {F_n} converges uniformly to F and  F'_n converges uniformly to G, then F' = G....
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