The copy of this question is posted [here]


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 $x=0$

**Note**:I have used this trick for approximation : $n! = e(-1)^{n+1}/(n+1) + O(1/n^2) (\mod \mathbb Z)$ implies that $\tan(n!)-n!=e((-1)^{n+1}/(n+1) + O(1/n^2) -1)(\mod \mathbb Z)$ 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 [![enter image description here][1]][1] 


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

[here]:
https://math.stackexchange.com/q/3700402/156150


  [1]: https://i.sstatic.net/SZIPZ.jpg