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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 $x=0$

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