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