Determine if $\displaystyle\sum_{k=1}^\infty\sin(kx)\sin\left(\cfrac{1}{kx}\right)$ is convergent Determine if the following series converges absolutely:
$$\sum_{k=1}^\infty \sin(kx)$$
and
$$\sum_{k=1}^\infty\sin(kx)\sin\left(\cfrac{1}{kx}\right)$$
I know how to deal with whether they converge. First one diverge by n-term test, second one converge by Dirichlet's Test. However, I do not know how to deal with whether they converge absolutely. Can someone teach me how to determine if both series converge absolutely. Thanks.

The first one, as it diverges, cannot converge absolutely (except for trivial case $x\in \pi\mathbb{Z}$). As for the second one - it is not absolutely convergent for most values of $x$ (when $x$ is not a rational multpile of $\pi$), the reason being roughly that in such case "many" values of $sin\ kx$ are "quite large", greater in absolute value than some fixed $0<a<1$. As the second term - $\sin \frac{1}{kx}$ - can be approximated by $\frac{1}{kx}$, which is divergent, it follows that partial sums are unbounded. Of course, it's just a vague sketch of a proof. You can find precise formulation of the above ideas and relevant results here
