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Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational?

This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an open problem? Are there any known results about it?

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    If your search $1.8798538621752585$ at ISC, you get only this sum and its variants. In particular, no rational value was found. But also no value in terms of other known quantities was found. ( http://isc.carma.newcastle.edu.au ) – Gerald Edgar Aug 04 '17 at 16:07
  • Can we not do something after which the proof of irrationality of $e$ can be imitated ? – Aditya Guha Roy Aug 05 '17 at 16:11
  • @adityaguharoy I attempted to do just that, folloowing the proof that $e$ is irrational. I got stuck, and that is what prompted me to pose this question on MSE. – Mark Fischler Apr 08 '19 at 18:55

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