Conditional convergence of a series involving $sin n \theta$

Question

I recently stumbled upon the series $$\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)\frac{\sin n \theta}{n}.$$ Consider all values of $\theta$ except $k \pi$ where $k$ is an integer. Is the series conditionally convergent? I know that $\sum_{n=1}^{\infty} \frac{\sin n \theta}{n}$ is convergent. But what about this?

Answer 1

Abel's Test implies that: If $a_n$ is monotonically decreasing and tends to $0$, then series $$\sum_{n=1}^{\infty}a_n\sin nx$$ is convergent for any $x\in \mathbb{R}$. Let $a_n=\frac{1}{n}\sum_{k=1}^{n}\frac{1}{k}$, you can get the idea！