sequences and series - Is this :$sum_{n=1}^{infty } frac{tan(frac{1}{n!})}{arctan (n!)}$ convergent sum?

How do I evaluate this sum :

$$\displaystyle\sum_{n=1}^{\infty } \ \frac{\tan(\frac{1}{n!})}{\arctan ({n!})}$$ if it is convergent ?.

Note: I think the limit of it's general term is $0$ as shown here in WA.

and i will surprised if it is convergent

Note: I edited the question beacuse i meant $arctan(n!)$ in the denominator

Answer

One has, as $n \to \infty$,

$$\frac{\tan \frac1{n!}}{\arctan (n!)}=\frac{\tan \frac1{n!}}{\frac{\pi}2-\arctan \frac1{n!}}\sim \frac2{\pi} \cdot \frac1{n!}$$ then by the comparison test the series $\displaystyle \sum_{n\ge1}\frac{\tan \frac1{n!}}{\arctan n!}$ is convergent.