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I was struck by the numerical coincidence $$ e + 2\pi = 9.0014671356... $$ which is within 0.02% of 9.

Is this just a coincidence, or can a sensible explanation of this be found, e.g., via a series expansion of some special function that has terms involving both $e$ and $2\pi$?

user1998586
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    Some related-but-difference numerical coincidences: https://math.stackexchange.com/questions/724872/why-is-e-pi-pi-so-close-to-20 https://math.stackexchange.com/questions/724872/why-is-e-pi-pi-so-close-to-20 https://xkcd.com/1047/ – user1998586 Feb 04 '22 at 17:56
  • Actually, $9.00014671356...$ is not really close to $9$. This here is really close to an integer, with $262 537 412 640 768 743.99999999999925...$. This has a good reason, but for your case I doubt this. – Dietrich Burde Feb 04 '22 at 17:59
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    I mean, $e$ is a little bit under $3$, $2\pi$ is a little bit over $6$.... – Randall Feb 04 '22 at 17:59
  • Ah, I see this has already been asked: https://math.stackexchange.com/questions/1711437/an-integral-for-2-pie-9 – user1998586 Feb 04 '22 at 18:04

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