$x\in (0,\pi)$ ,Prove that: \begin{align} \sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3 \end{align}
the inequality holds for all integer $n$
I tried Fourier, or Dirichlet kernel, but they don't work.Thanks for your attention!
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Please state where this problem from. – Nathaniel Bubis Aug 02 '12 at 12:30
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while proving the Fejér-Jackson inequality, from http://www.artofproblemsolving.com/Forum/viewtopic.php?t=114058 5 floor – Golbez Aug 02 '12 at 12:32
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This left hand side is simply the Fourier series of a Sawtooth wave. All you now have to do is prove that the polynomial to the right is smaller then the straight line.
Nathaniel Bubis
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How to compare the partial sum of the Fourier series with a polynomial? – Golbez Aug 02 '12 at 14:14
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@Golbez - You may find this helpfull as well: http://math.stackexchange.com/questions/57054/asymptotic-error-of-fourier-series-partial-sum-of-sawtooth-function – Nathaniel Bubis Aug 02 '12 at 15:20