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 tetration limit ?? sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 06/11/2015, 10:27 AM (This post was last modified: 06/14/2015, 10:08 AM by sheldonison.) (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). $\eta=\exp(1/e)\;\;k=\ln(\ln(\eta+\frac{1}{n}))+1\approx \frac{e}{\eta \cdot n} \approx \frac{1.8816}{n}\;\;\;$ Now we have switched it to a problem of iterating $z \mapsto \exp(z)-1+k\;$. 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. $x=\sqrt{-2k}\;\;\; L = x - \frac{x^2}{6}+\frac{x^3}{36}+...\;\;\approx \sqrt{ \frac{-2e}{\eta \cdot n}}\;\;\;\text{period}=\frac{2\pi i}{L}\;\;\$ this is the period at the fixed point. $\Re(\text{period}) = \Re(\frac{2\pi i}{L}) \approx 2\pi \sqrt{\frac{\eta \cdot n}{2e}} \approx \sqrt{10.49n}\;\;\;$ 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 $\lim_{n \to \infty} \text{sexp}_{(\eta+1/n)}\left[\Re(\text{period})-2\right] =k \;\;\; k\approx 388.787398293917704779$, where the period is from the equation above. The correct middle term is probably $\text{slog}_e(\frac{n}{e}-1)\;\;$ Note that $\text{slog}_e(n/e-1)=\text{slog}_e(\eta^{\eta^n})-2\;\;$ So then, we have the following conjectured equation, where I'm pretty sure a 1-cyclic theta is required as n gets arbitrarily large, $\theta(\text{slog}_e(n/e-1))$, whose predicted amplitude is probably about +/-0.002 $\lim\limits_{n\to\infty} \text{sexp}_{(\eta+1/n)}\left[2\pi\sqrt{\frac{\eta\cdot n}{2e}} + \text{slog}_e(\frac{n}{e}-1) + C + \mathcal{O}(\theta) \right] -n = 0\;\;\;\;C \approx -2 - \text{slog}_e(388.7874/e-1)$ - Sheldon « Next Oldest | Next Newest »

 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 Updated results for tetration limit - by sheldonison - 10/31/2010, 03:32 PM 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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