Is the series $ \sum u_{n}$
$ u_{n}=n!\prod_{k=1}^n \sin\left(\frac{x}{k}\right)$
$ x\in]0,\pi/2] $
convergent or divergent?
We have:
$ u_{n}\leq n!\prod_{k=1}^n \frac{x}{k}$
$ u_{n}\leq x^n$
If $0
$ u_{n}\geq n! \prod_{k=1}^n \frac{2x}{\pi k}$
$ u_{n} \geq \prod_{k=1}^n \frac{2x}{\pi}$
If $x=\pi/2$, $u_{n}\geq1$, $\sum u_{n}$ is divergent.
What about the case $x\in[1,\pi/2[$ ?