Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally? I think that the series $$\sum_{n=1}^{\infty} \dfrac{\sin(n)}{n}$$ converges conditionally, but I´m not able to prove it. Any suggestions ?

Using Fourier series calculations it follows 
$$
\sum_{n=1}^{\infty}\frac{\sin(n x)}{n}=\frac{\pi-x}{2}
$$
for every $x\in(-\pi,\pi)$. Your sum is $\frac{\pi-1}{2}$.
