How to find the values of $\theta$ for which the series $$\sum_{n=1} ^{\infty} \frac{(1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ ...+ \frac{1}{n})}{n} \cos n\theta$$ is convergent?
What I could show is that $$\sum_{n=1} ^{\infty} \frac{\cos n\theta}{n} $$ is convergent where $\theta$ is not any integer multiple of $2\pi$.
Any help would be appreciated. Thanks in advance.
Just like $\sum_{n=1} ^{\infty}(\cos n\theta)/n$, we can show that the series is convergent when $\theta$ is not an integer multiple of $2\pi$.
Indeed, if $\theta$ is not an integer multiple of $2\pi$, we have $$\left|\sum\limits_{n=1}^N \cos n\theta\right|=\left|\frac{\sin(N+\frac{1}{2})\theta-\sin \frac12\theta}{2\sin \frac12\theta}\right|\le \frac1{|\sin \tfrac12\theta|}$$ and $$\frac{1}{n}\sum\limits_{k=1}^n\frac{1}{k}=\frac{\log n}{n}+o(1),\text{ when } n\to\infty.$$ Thus the series $$\sum_{n=1} ^{\infty} \frac1n\left(1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ ...+ \frac{1}{n}\right) \cos n\theta$$ is convergent by the Dirichlet criterion.
On the other hand, if $\theta=2k\pi$, with $k\in\Bbb Z$, the series becomes $$\sum_{n=1} ^{\infty} \frac1n\left(1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ ...+ \frac{1}{n}\right),$$ which is divergent.
