Sum $\sum_{n=1}^{\infty}(1+1/2+...+1/n)\frac{\sin{nx}}{n}$ convergent or divergent?
For what values of $a$ does $\sum_{n=1}^\infty \left( 1+\frac12 + \dotsb + \frac1n \right) \frac{\sin (na)}{n}$ converge?