TITLE: How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?
QUESTION [6 upvotes]: I proposed this question on MSE but some comments affirmed that is  unsolved problem and no answer. I would like to see what MO say about it.
How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$
Note : I used many criterions of convergence to test whether it converges but i didn't succeed.
Thank you for any help .

REPLY [5 votes]: I computed $f(n)=\sum_{k=0}^n \tan(k!)$ for different $n$,
and got the following plot. It does not seem to have a limit.

Mathematica code:
Block[{$MaxExtraPrecision = 600},
 lst = Table[Tan[n!], {n, 0, 200}];
 ListPlot[N[Accumulate[lst], 200], PlotRange -> All]
 ]
{"set_name": "stack_exchange", "score": 6, "question_id": 209990}