Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite Pentation (and x-srt-x)
#12
So this is my attempt at a proof, and there are a bunch of inequalities, which always confuse me Tongue, so let me know if there are any mistakes.

Let as usual. I will use k (instead of n) to avoid confusion with .

Lemma 1 (Knoebel). iff for all .

Lemma 2. for all real .
Proof.
The base- tetrational function is continuous and monotonic?

Lemma 3. for all positive real and real .
Proof.
The function for all , thus . If , then , so .

Lemma 4 (lower bound). For all integer , .
Proof.
Since , then obviously . Together with lemma (2), this implies that . Substituting (hypothesis), this can be written as which implies by lemma (1).

Lemma 5 (decreasing). For all integer and , .
Proof.
We have for all integer by definition. It follows that
by lemma (3). Thus which implies by lemma (1).

Theorem.
Proof.
for all by lemma (4) and lemma (5).
... In the limit, the squeeze theorem and completeness should guarantee that the limit exists and converges to .

Is this right?

Andrew Robbins
Reply


Messages In This Thread
Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 09:16 AM
RE: Infinite Pentation (and x-srt-x) - by andydude - 04/12/2009, 08:58 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Improved infinite composition method tommy1729 2 24 06/14/2021, 04:17 AM
Last Post: JmsNxn
  pentation and hexation sheldonison 9 16,337 09/18/2019, 02:34 PM
Last Post: sheldonison
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 2,564 03/20/2018, 12:16 AM
Last Post: tommy1729
  Tetration is pentation. This deserve more thinking. marraco 2 5,900 03/30/2015, 02:54 PM
Last Post: marraco
  [2015] 4th Zeration from base change pentation tommy1729 5 9,936 03/29/2015, 05:47 PM
Last Post: tommy1729
  Mizugadro, pentation, Book Kouznetsov 41 77,954 03/02/2015, 08:13 PM
Last Post: sheldonison
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 18,817 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 8,256 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 10,959 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 6,180 09/06/2012, 02:01 AM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)