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$x\in (0,\pi)$ ,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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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.

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  • $\begingroup$ How to compare the partial sum of the Fourier series with a polynomial? $\endgroup$ – Golbez Aug 2 '12 at 14:14
  • $\begingroup$ @Golbez - You may find this helpfull as well: math.stackexchange.com/questions/57054/… $\endgroup$ – nbubis Aug 2 '12 at 15:20

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