Why do computer programs (Wolfram Alpha, Symbolab) think $\sum_{n=0}^{\infty} \sin(\frac{\pi}{2}n!)$ is divergent? I've been recently looking at the series
$$\sum_{n=0}^{\infty} \sin(\frac{\pi}{2}n!)=1+1+0+0+0+0+0+\cdots,$$
which should equal $2$. However, programs such as Wolfram Alpha, Symbolab, etc., tell me that this series is divergent. Can someone explain what's happening?

For $n \ge 2$ we have
$\frac{\pi}{2}n!= \pi k_n$ with $k_n \in \mathbb N$. Hence:
$\sin(\frac{\pi}{2}n!)=0$ for $n \ge 2$ and the series is convergent.
