I need to study where the following series is convergent: $$\displaystyle \sum_{n=1}^{\infty} \frac{\{ n \alpha\}}{n}$$ Here $\{x\}$ denotes the fractional part of $x$ and $\alpha \in \mathbb{R}_{>0}$. I've already proven that it converges if $\alpha \in \mathbb{N}$ and that it diverges if $\alpha \in \mathbb{Q} \ \backslash \ \mathbb{N}$ (It is done rather easily, namely by comparison with the Harmonic Series).
What should I do when $\alpha$ is irrational?
For $\alpha\not \in \Bbb{Z}$,
$\sum_{n\ge 1} \frac{\{n\alpha\}}n$ diverges because one of $\{n\alpha\}, \{(n+1)\alpha\}$ is larger than $\min(\{\alpha\},1-\{\alpha\})$.