Studying convergence of the series involving fractional part
Solution 1:
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 $\min(\{\alpha\},1-\{\alpha\})$.
Studying convergence of the series involving fractional part
Solution 1:
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 $\min(\{\alpha\},1-\{\alpha\})$.