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Math Help - a problem about fourier series

  1. #1
    Junior Member
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    a problem about fourier series

    Show that the trigonometric series


    ∑1/log(n)*sin(nx) n from 2 to infinity

    converges for every x, yet it is not the fourier series of a riemann integrable
    function.


    thanks
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Use Abel's test to show that the complex power series \sum (1/\ln n)z^n converges on the unit circle (and therefore so does its imaginary part \sum (1/\ln n)\sin nx).

    I'm not sure how to do the second part of the question. If this series were the Fourier series of a square-integrable function then you could use the Parseval theorem to say that \sum(1/\ln n)^2 converges (which it doesn't). But if you are only allowed to assume that the function is integrable then this idea won't work.
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