Does $\sum_{k=1}^\infty(1+\frac{1}{2}+ \dots +\frac{1}{k})\frac{\sin kx}{k}$ converge absolutely? For $x\neq k\pi,k\in\mathbb{Z}$,does the following series converge absolutely?
$$\sum_{k=1}^\infty(1+\frac{1}{2}+ \dots +\frac{1}{k})\frac{\sin kx}{k}$$

Helpful Observation: note that
$$
1 + \frac 12 + \cdots + \frac 1k=\ln(k)+\gamma + o(1)
$$
Where $\gamma$ is a constant (the Euler-Mascheroni constant, in fact) and $o(1)$ is some sequence converging to zero.

The answer to your question is that the sequence converges conditionally, but not absolutely. It is possible to show conditional convergence using the Dirichlet test and show absolute divergence with a comparison test.
