How can I show that the following series is convergent or divergent ?
$$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$
I want to use Abel-Dirichlet criteria. I've already noticed that $1 \over n$ decreases to zero, and is $>0$ for any $x$ bigger than $1$. The problem is that I do not know how to show that $\color{red}{\sum_{n=1}^{\infty}\cos{(n)} \sin{(nx)}}$ has a bounded sequence of partial sums. Can you please help me? While trying to show the boundedness of this series I used $\color{red}{\sin(x+y)-\sin(x-y)=2\cos (x) \sin (y)}$ formula, but could not get anything.
It is the case to notice that: $$ \cos n \sin(nx) = \frac{1}{2}\left(\sin(n(x+1))+\sin(n(x-1))\right),$$ and since for any $z\in(0,2\pi)$ we have: $$ \sum_{n\geq 1}\frac{\sin(nz)}{n}=\frac{\pi-z}{2}, $$ it follows that: $$ \sum_{n\geq 1}\frac{\cos n}{n}\,\sin(nx)$$ is a $2\pi$-periodic function, discontinuous in $x=1$ and $x=2\pi-1$, elsewhere differentiable with derivative $-1$:
Since such a function vanishes for $x\in \pi\mathbb{Z}$, we have, for any $x$: $$\left|\sum_{n\geq 1}\frac{\cos n}{n}\,\sin(nx)\right|\leq\color{red}{\pi-1}.$$