Talk:List of representations of e

Is there any BBP-type series for e?

Many mathematical constants have Bailey–Borwein–Plouffe formula for series. These allow one to calculate an arbitrary digit of the constant without having to previously calculate all the prior digits. Our articles include such series for pi#Spigot algorithms and the natural logarithm of 2#Series representations. So I looked for one for e, hoping that I could add it to this article. I found this paper which seems to indicate that no such series is known for e and even hints that it may not be possible to get one for e. JRSpriggs (talk) 02:03, 21 May 2016 (UTC)

e as the number of arrangements by number derangements

My addition $e=\lim _{n\to \infty }{n! \over !n}$ was reverted. This is supposed to be a list of representations of e, why is this representation not worthy of being added to the list? User:Deacon Vorbis said it's because it's just the reciprocal of an identity for $e^{-1}$, but this applies to all identities in this list plus there isn't a page List of representations of 1/e, so this point is kinda moot. --FUZxxl (talk) 21:41, 12 August 2017 (UTC)

Ah, I was in the middle of typing my response, but you beat me to the talk page. To clarify the revert I made to the addition of $\lim _{n\to \infty }{\frac {n!}{!n}},$ the denominator in this expression is simply

$!n=n!\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.$

Thus, after cancelling the $n!,$ what's left is simply 1 over the Maclaurin series for 1/e, which is already listed in the article. --Deacon Vorbis (talk) 21:48, 12 August 2017 (UTC)

That's not really a good argument -- all these representations can be translated into one-another by means of algebraic manipulations. So why should this one not belong? After all, it shows an interesting connection between combinatorics and Euler's number, while appearing some times in practice. So at last, let me ask, what are the objective standards for an entry to belong into this list? --FUZxxl (talk) 21:51, 12 August 2017 (UTC)

Many of the representations listed require derivations far beyond simple algebraic manipulation, whereas the one you wanted to add reduced to the expression for 1/e after one cancellation after writing out the standard expression for !n. As far as standards, I doubt it's worth it to try to set anything out explicitly. But common sense goes a long way here, and there's no need to clutter up the article with nearly redundant expressions. (I think that a few already on here could probably be trimmed out too). --Deacon Vorbis (talk) 22:13, 12 August 2017 (UTC)

You have not answered my question. What are the objective standards for an entry to belong into this list? Note that without knowing the sum formula for $!n$, it can be quite hard to see how this limit relates to $e$. Look at it from the perspective of someone who just found that $n!/!n$ converges to a number apparently close to $e$, the first thing you would do is look at this article and check if it indeed converges to $e$, making this entry fairly useful to the reader. I still don't see why it should be left out other than your very arbitrary opinion. --FUZxxl (talk) 22:21, 12 August 2017 (UTC)

I'm not sure how much clearer I can make:

As far as standards, I doubt it's worth it to try to set anything out explicitly. But common sense goes a long way here, and there's no need to clutter up the article with nearly redundant expressions.

But if you do know the formula for !n, the relationship to e is pretty easy, and that one is included here. The target of a search for further information should lead someone to derangements (and not to here), which does have more information along these lines. --Deacon Vorbis (talk) 00:24, 13 August 2017 (UTC)

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Conjectures

I note that there is a proof on the arxiv for the [https://wiki.riteme.site/wiki/List_of_representations_of_e#Conjectures conjectured continued fraction expansion]. A brief play with the numbers suggests that the difference between successive approximations is equivalent to the 'consideration of upper bounds' infinite series.

Icecolbeveridge (talk) 16:29, 4 August 2020 (UTC)

Ehh, a random arXiv posting isn't really that usable. Poking around, I even saw a much simpler one-liner (a few more with the steps worked out) on MO. Given that the source is just itself a preprint as far as I can tell and only generates conjectures, I'd say it's probably best to just remove it unless something better comes along. –Deacon Vorbis (carbon • videos) 16:46, 4 August 2020 (UTC)