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I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{ n! \bmod 2\pi\mathbb Z\}$ is dense in $[0,2\pi]$ then $\tan(n!)$ will be large arbitrarily infinitely often, and the difference $\tan(n!)-n!$ would be probably small. However I don't know anything about limit of the fraction $\frac{\tan(n!)}{n!}$ for large $n$, And I have used Stiriling formula but it weren't helpful, This really forbide me to determine how $x$ values should be to get the titled series converge, then my question is :

Question: For which values of real number $x$ we have $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges ? We may avoid trivial case $x=0$ , In particular how we prove the set of values $\{ n! \bmod 2\pi\mathbb Z\}$ is dense in $[0,2\pi]$ ?

Note:I have used this trick for approximation : $n! = e(-1)^{n+1}/(n+1) + O(1/n^2) (\mod \mathbb Z)$ implies that $\tan(\pi n!)-n!=\pi e((-1)^{n+1}/(n+1) + O(1/n^2)) -\frac{1}{\pi})(\mod \mathbb Z)$ but this what could saying to me ? Probably conditional convergence could be happen here .

Addedendum

I'm interested to evaluate the above series because this need to prove $\{ n! \bmod 2\pi\mathbb Z\}$ is dense in $[0,2\pi]$ which is related to irrationality measure and Bounds approximation of $\pi$ which it is interesting in number theory and in the same time looking to the behavior of $\tan(n!)-n!$ for large $n$ which it help to get bounds for some arithmitic functions , Also to loook to the relationship of convergence of series to lacunar Fourier series which we have a lot of results on convergence for them

using mathematica code for $n=400$, $\tan(n!)-n!$ I have got the below plot and this is the code ,

Block[{$MaxExtraPrecision = 800},
  lst = Table[Tan[n!] - n!, {n, 0, 400}];
  ListPlot[N[Accumulate[lst], 200], PlotRange -> All]
  ]

enter image description here

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    $\begingroup$ Does this question fit into a larger picture? Why is this question important to you? $\endgroup$
    – Ben McKay
    Commented Jun 2, 2020 at 7:13
  • $\begingroup$ @BenMcKay, Sorry I missed to add motivation of that question, I'm already adding that in Addedendum $\endgroup$
    – zeraoulia rafik
    Commented Jun 2, 2020 at 12:25
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    $\begingroup$ However , I don't know why downvote however that question is in the level of research and in the same time I have showed its motivation $\endgroup$
    – zeraoulia rafik
    Commented Jun 2, 2020 at 18:22
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    $\begingroup$ Your question is intersting whene i plot the function $f(t)=\tan(t)-t$, we can see that almost values of $f(n)$ with $n$ positive integers are negative and diverge fast. $\endgroup$
    – Lagrida Yassine
    Commented Jun 2, 2020 at 20:29
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    $\begingroup$ 12 versions in 12 hours. $\endgroup$
    – Gerry Myerson
    Commented Jun 3, 2020 at 0:18

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