Is there any closed formula for the series

$$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k}$$

or for the sum

$$\sum_{k=1}^{n}\frac{\sin (kx)}{k}$$

where $x$ is an real number. Thank you.

  • 2
    $\begingroup$ $$\sum_{k\geq 1}\frac{\sin k}{k}=\frac{\pi-1}{2}$$ by the pointwise convergence of the Fourier series of $\frac{\pi-x}{2}$ (defined on $(0,2\pi)$ and extended by periodicity). Convergence is not uniform due to Gibbs phenomenon. $\endgroup$ – Jack D'Aurizio Apr 5 '18 at 12:15
  • $\begingroup$ Sees an open shop. Enters the shop. Takes some products. Leaves the shop. $\endgroup$ – Did Apr 5 '18 at 12:26
  • $\begingroup$ A mathematician enters in a coffee. Splash. $\endgroup$ – Jack D'Aurizio Apr 5 '18 at 12:59

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