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 $x=0$

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