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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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    while proving the Fejér-Jackson inequality, from http://www.artofproblemsolving.com/Forum/viewtopic.php?t=114058 5 floor2012-08-02

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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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    @Golbez - You may find this helpfull as well: http://math.stackexchange.com/questions/57054/asymptotic-error-of-fourier-series-partial-sum-of-sawtooth-function2012-08-02