Does limit of $\tan(n !)-n!,n\to \infty$ exists ?And what is its upper bound?

Question

I have tried many times to show if $\tan(n !)-n!$ exist or not but I can't , I have used this idea $n! = e(-1)^{n+1}/(n+1) + O(1/n^2) (\mod \mathbb Z)$ implies that $\tan( n!)-n!= e((-1)^{n+1}/(n+1) + O(1/n^2)) -1)(\mod \mathbb Z)$ to get such bounds of that sequence but it doesn' seems clear to me whethere it is converge or diverge , Then my question here is :Does limit of $\tan(n !)-n!$ exists ?And what is its upper bound? Wolfram alpha dosn't give any result only series asymptotic at infty as shown here

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]
  ]

Edit: Probabliy from density $n ! \bmod \mathbb{Z}$ the sequence would be diverge we may add this partial question :Does it will be diverge to $-\infty$ or just diverge ?

Related question:

https://mathoverflow.net/q/361945/51189