Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
proof: Limit of self-super-roots is e^1/e. TPID 6
#1
In reply to
http://math.eretrandre.org/tetrationforu...73#pid4073

First it is easy to see that for :

( is the lower fixed point of )

Hence for we have for all :
(*)

We also know that for , quite fast, particularly for each there is an such that for all :
(**) .

Now we lead proof by contradiction, suppose that
where .

Then there must be a subsequence and such that this subsequence stays always more than apart from :
.
I.e. there is and such that
either or .

By (*) and (**) we have such that for all :
and .
As is monotone increasing for we have also
and .

This particularly means and hence none of the can be the self superroot, in contradiction to our assumption.
Reply
#2
Wow! Very nice! You make it seem so easy. Smile I've been working on that one for while, ever since the xsrtx thread.
Reply
#3
The same method of proof could possibly be used to easily prove that, possibly for all k>4, limit of self-hyper-k-root(x) as x -> infinity = (defined as the largest real x such that , i.e. where the maximum of self-hyper-(k-1)-root function occurs; let's establish this notation); yeah I know, I only substituted the pentation-analogues into the proof and quickly checked.
Reply
#4
(07/10/2010, 05:19 AM)Base-Acid Tetration Wrote: The same method of proof could possibly be used to easily prove that, possibly for all k>4

The thing is: to define the hyper k-self-root you need a hyper (k-1) operation defined on the reals.
And we still have several methods of doing this without equality proofs.
Reply


Possibly Related Threads…
Thread Author Replies Views Last Post
  On extension to "other" iteration roots Leo.W 21 2,274 Yesterday, 07:47 AM
Last Post: bo198214
  RED ALERT : TPID CONJECTURES GONE ??? tommy1729 4 66 08/12/2022, 10:08 PM
Last Post: tommy1729
Question A Limit Involving 2sinh Catullus 0 87 07/17/2022, 06:15 AM
Last Post: Catullus
Question TPID 6 Catullus 1 131 07/04/2022, 12:55 PM
Last Post: tommy1729
  TPID 4 tommy1729 30 52,940 06/18/2022, 10:40 PM
Last Post: tommy1729
  Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 12,028 05/26/2021, 11:55 PM
Last Post: MphLee
  Is bugs or features for fatou.gp super-logarithm? Ember Edison 10 16,818 08/07/2019, 02:44 AM
Last Post: Ember Edison
  Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 17,788 06/10/2019, 04:29 AM
Last Post: Ember Edison
  Where is the proof of a generalized integral for integer heights? Chenjesu 2 4,978 03/03/2019, 08:55 AM
Last Post: Chenjesu
  Inverse super-composition Xorter 11 26,871 05/26/2018, 12:00 AM
Last Post: Xorter



Users browsing this thread: 1 Guest(s)