Let $\alpha$ be an irrational number, and $\left\{ \cdot \right\}$ denotes the fractional part function. We have focused on how $\left\{ n! \alpha \right\}$ distributes in this MO question. And now I am interested in the series $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$.

The above question mentions that, I quote, there is a subset of $\mathbb{R}$ of Hausdorff dimension $1$, such that the corresponding sequence is bounded away from $0$, which means the series $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$ must diverge.

Though we cannot tell the exact distribution of $\left\{ n! \alpha \right\}$ yet, it might be reasonable to suspect that the convergence of $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$ have been fully settled, just as the series $\sum_{n = 1}^\infty \sin( n! \pi \alpha )$.

Sadly, I exhaust my scope of search without finding a similar result. As Hanson points out, consider a specific number, such as $\pi$, would be unbearably hard. Therefore, I want to ask if there is an $\alpha$ such that the series converges, and if there is some approach to describe these $\alpha$. (In particular, the canonical choice $\alpha = \mathrm{e}, \frac{1}{\mathrm{e}}$ fails to work.) Moreover, I want to request for references to this series.

Any advice is welcomed.