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tetration limit ??
#40
(05/28/2015, 11:32 PM)tommy1729 Wrote: Let n be a positive integer going to +oo.

lim [e^{1/e} + 1/n]^^[(10 n)^{1/2} + n^{A(n)} + C + o(1)] - n = 0.

Where C is a constant.

Conjecture : lim A(n) = 1/e.

Its a curious equation. I viewed it from a different angle: What is the slog_{1/e+1/n}(n)? But I couldn't figure out why you were interested in slog(n) as opposed to say, slog(e^e) or something like that that made more sense to me. e^e is the cusp of where this tetration function takes off, and the function starts growing superexponentially. But the (1/n) means it might take 1 or 2 more iterations to reach (1/n), Or if n is hyperexponentially large = sexp(4.5), then 3 extra iterations. But most of the time is spent getting to e^e. And that equation is dominated by approximately real(Pseudo period)-2. And you included an O(1) term in your equation anyway, which implies C isn't an exact constant.

So then my counter conjecture would be that lim sexp_{1/e+1/n)(real(Period)-2)=constant, and that constant seems to be about 388 as n goes to infinity. But that seemed to be a very different equation than the one you had in mind, so I thought it would be off topic, so I didn't mention it. But yeah, I have equations for the pseudo period, which I posted below.

Then there is your approximation itself. slog_{1/e+1/n}(n) = (10n)^{1/2} + n^{A(n)} + C. Can you explain why you think this is the right approach or equation? It doesn't seem to match the approximation I have for real(pseudo_period)-2...

The equations for the fixed point and Period are approximately as follows. One can see that the resulting period has a sqrt term, but not sqrt(10n).


Now we have switched it to a problem of iterating . In the limit, the fixed point goes to zero. This iteration mapping has a simpler Taylor series for the fixed point L, from which we can generate the Pseudo Period.

this is the period at the fixed point.

this is close to sqrt(10n). But I don't understand your n^A(n)~=n^(1/e) term; [(10 n)^{1/2} + n^{A(n)}].

Anyway, my counter-conjecture is that
, where the period is from the equation above.

The correct middle term is probably Note that So then, we have the following conjectured equation, where I'm pretty sure a 1-cyclic theta is required as n gets arbitrarily large, , whose predicted amplitude is probably about +/-0.002



- Sheldon
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Messages In This Thread
tetration limit ?? - by tommy1729 - 04/01/2009, 05:49 PM
RE: tetration limit ?? - by nuninho1980 - 04/01/2009, 08:15 PM
RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:58 PM
RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 12:53 AM
RE: tetration limit ?? - by bo198214 - 04/03/2009, 12:49 PM
RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 05:54 PM
RE: tetration limit ?? - by bo198214 - 04/02/2009, 02:50 PM
RE: tetration limit ?? - by tommy1729 - 04/02/2009, 09:24 PM
RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:56 PM
RE: tetration limit ?? - by tommy1729 - 04/02/2009, 10:39 PM
RE: tetration limit ?? - by tommy1729 - 05/29/2011, 07:28 PM
RE: tetration limit ?? - by bo198214 - 05/31/2011, 10:34 AM
RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 06:12 PM
RE: tetration limit ?? - by bo198214 - 04/06/2009, 10:49 PM
RE: tetration limit ?? - by nuninho1980 - 04/07/2009, 01:35 AM
RE: tetration limit ?? - by nuninho1980 - 04/04/2009, 02:21 PM
RE: tetration limit ?? - by gent999 - 04/14/2009, 10:12 PM
RE: tetration limit ?? - by bo198214 - 04/14/2009, 10:31 PM
RE: tetration limit ?? - by gent999 - 04/15/2009, 12:18 AM
RE: tetration limit ?? - by bo198214 - 04/15/2009, 01:35 PM
RE: tetration limit ?? - by tommy1729 - 04/15/2009, 03:05 PM
RE: tetration limit ?? - by gent999 - 04/15/2009, 04:41 PM
RE: tetration limit ?? - by tommy1729 - 04/29/2009, 01:08 PM
RE: tetration limit ?? - by BenStandeven - 04/30/2009, 11:29 PM
RE: tetration limit ?? - by tommy1729 - 04/30/2009, 11:38 PM
RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:35 AM
RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:00 AM
RE: tetration limit ?? - by JmsNxn - 04/14/2011, 08:17 PM
RE: tetration limit ?? - by tommy1729 - 05/28/2011, 12:28 PM
RE: tetration limit ?? - by nuninho1980 - 10/31/2010, 10:31 PM
RE: tetration limit ?? - by JmsNxn - 05/29/2011, 02:06 AM
RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:29 PM
RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:33 PM
RE: tetration limit ?? - by tommy1729 - 05/28/2015, 11:32 PM
RE: tetration limit ?? - by sheldonison - 06/11/2015, 10:27 AM
RE: tetration limit ?? - by sheldonison - 06/15/2015, 01:00 AM
RE: tetration limit ?? - by tommy1729 - 06/01/2015, 02:04 AM
RE: tetration limit ?? - by tommy1729 - 06/11/2015, 08:25 AM

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