As mentioned in the title.
I’m trying to solve this problem by estimating $1+1/2+…+1/n$:$$1+\frac{1}{2}+…\frac{1}{n}=\ln(n)+\gamma+o(1)$$ Where $\gamma=0.57721...$ is Euler–Mascheroni constant. Then we get $$\sum_{n=1}^\infty (1+\frac{1}{2}+…\frac{1}{n})\frac{\sin{nx}}{n}= \sum_{n=1}^\infty \frac{\ln n\sin{nx}}{n}+ \gamma\sum_{n=1}^\infty \frac{\sin{nx}}{n}+ \sum_{n=1}^\infty \frac{\sin{nx}}{n} o(1)$$
I have problem continuing because I do not know how to handle the first term and the third term of RHS.
P.s. I already know that $\sum_{n=1}^\infty \frac{\sin{nx}}{n}$ is convergent due to Dirichlet’s test.