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