# Finding closed forms for a trigonometric series or for the partial sums

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.

• $$\sum_{k\geq 1}\frac{\sin k}{k}=\frac{\pi-1}{2}$$ 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. – Jack D'Aurizio Apr 5 '18 at 12:15
• Sees an open shop. Enters the shop. Takes some products. Leaves the shop. – Did Apr 5 '18 at 12:26
• A mathematician enters in a coffee. Splash. – Jack D'Aurizio Apr 5 '18 at 12:59