Tetration Asymptotic Series
#1
Question 
Sqrt(2) ^^ x is approximated better and better by 2 - a*ln(2)^x + (a^2/(4*(1 - 1/ln(2))))*ln(2)^(2*x), where a is lim x → ∞ (2 - sqrt(2) ^^ x) / ln(2) ^ x. It has an error term of big O of ln(2) ^ 3x. Does anyone know a formula for the asymptotic approximations of a ^^ x when 1 < a < η? Or even more terms, but not a formula for them?
All of the higher order terms for b^^x are expressible, in closed form in terms of b's version of a. For base b the analog of a is lim x → ∞ (LambertW(-ln(b))/ln(b)-b^^x)/LambertW(-ln(a))^b.
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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Messages In This Thread
Tetration Asymptotic Series - by Catullus - 06/08/2022, 05:47 AM
RE: Tetration Asymptotic Series - by tommy1729 - 06/08/2022, 12:18 PM
RE: Tetration Asymptotic Series - by Catullus - 06/09/2022, 04:49 AM
RE: Tetration Asymptotic Series - by bo198214 - 06/09/2022, 05:58 PM
RE: Tetration Asymptotic Series - by Catullus - 06/09/2022, 09:25 PM
RE: Tetration Asymptotic Series - by bo198214 - 06/09/2022, 09:34 PM
RE: Tetration Asymptotic Series - by JmsNxn - 06/10/2022, 11:27 PM
RE: Tetration Asymptotic Series - by bo198214 - 07/02/2022, 10:37 AM
RE: Tetration Asymptotic Series - by JmsNxn - 07/03/2022, 07:45 AM
RE: Tetration Asymptotic Series - by Catullus - 07/03/2022, 09:20 AM
RE: Tetration Asymptotic Series - by Gottfried - 07/03/2022, 10:18 AM
RE: Tetration Asymptotic Series - by Catullus - 06/10/2022, 09:56 AM
RE: Tetration Asymptotic Series - by JmsNxn - 06/10/2022, 08:52 PM
RE: Tetration Asymptotic Series - by Catullus - 06/10/2022, 10:50 PM
RE: Tetration Asymptotic Series - by JmsNxn - 06/10/2022, 11:01 PM
RE: Tetration Asymptotic Series - by Catullus - 07/04/2022, 11:19 PM
RE: Tetration Asymptotic Series - by JmsNxn - 07/04/2022, 11:55 PM
RE: Tetration Asymptotic Series - by Catullus - 07/05/2022, 01:19 AM
RE: Tetration Asymptotic Series - by JmsNxn - 07/05/2022, 01:29 AM

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