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As the title says, I'm wondering whether there is any known closed-from for the following series:

$$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$

Here I don't mean the double factorial (treated here) when I'm writing $(k!)!$, but the factorial of the integer $k!$.

Trying on WolframAlpha, I get the funny value $2.501388888888888888888890500626459...$, but no closed-form. If you have any reference which focuses on this number, I would also be interested. I found nothing so far.

Thank you very much!

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Watson
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    The fifth term is about $1.6 * 10^{-24}$ and the sixth term is much smaller than $10^{-133}$. Fast convergence indeed. :D – Burrrrb Jul 05 '16 at 19:51
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    It's highly unlikely, for a slightly curious reason: this number is not just transcendental but a form of Liouville number, and such highly approximable numbers tend not to be 'natural' - most of the numbers that we come across on a regular basis have (or are strongly believed to have) finite irrationality measure. – Steven Stadnicki Jul 05 '16 at 19:52
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    $\sum_{k\geq0}1/(k!)!\approx\sum_{k=0}^{3} 1/(k!)!=1801/720=2.5013\bar{8}$ – parsiad Jul 05 '16 at 19:53
  • https://oeis.org/A336686 – Mason Jul 26 '22 at 22:25

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