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How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$

Note : I used many criterions of convergence to show if it converges but i

didn't up.

Thank you for any help .

Edit: Also asked at MathOverflow

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Zeraoulia rafik
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    I see no reason to think this would converge. Is there a reason you'd think it might? – Thomas Andrews Jun 22 '15 at 23:45
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    I have no idea, but it may help, maybe one of you will have a flash of genius the series of $$\sin^p\left(n!\pi e\right) $$ is convergent for $p\ge 1$ – ParaH2 Jun 23 '15 at 01:20
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    Turns out this is an unsolved problem. – Lynn Jun 23 '15 at 01:38
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    The length of the comment chain raised a system flag. I trimmed most of the comments. Zeraoulia rafik, if you ask the same question on many SE sites, please cross link them (and only do that in very exceptional circumstances). – Jyrki Lahtonen Jun 24 '15 at 20:39
  • @JyrkiLahtonen, i asked the above question in MO after asking it here in SE – Zeraoulia rafik Jun 26 '15 at 00:12
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    I did see that you linked to this question in MO. You could have also edited this version at that time. – Jyrki Lahtonen Jun 26 '15 at 05:27
  • In order for that to converge, you would need $n!$ to be a very good approximation of multiples of $\pi$, which I suppose is possible but highly unlikely and I expect someone would have noticed if it were true. – Dark Malthorp Jun 29 '18 at 21:47

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