:confused: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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- Sep 20th 2007, 09:36 PMXingyuana problem about fourier series
:confused: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 - Sep 21st 2007, 12:23 AMOpalg
Use Abel's test to show that the complex power series $\displaystyle \sum (1/\ln n)z^n$ converges on the unit circle (and therefore so does its imaginary part $\displaystyle \sum (1/\ln n)\sin nx$).

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 $\displaystyle \sum(1/\ln n)^2$ converges (which it doesn't). But if you are only allowed to assume that the function is integrable then this idea won't work.