Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Arithmetic in the height-parameter (sums, series)
#1
Hi -

using the notation and we do arithmetic in the height (or "iteration") parameter like

What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get


For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.

So, with base b=sqrt(2) the following expression



seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.

On the other hand, the analytical continuation for the geometric series with constant quotient q at q=2 gives



But -substitued this into the height-parameter- then we should also have



where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).

---

In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:




Just another plot of meditations... <phew>

Gottfried
Gottfried Helms, Kassel
Reply


Messages In This Thread
Arithmetic in the height-parameter (sums, series) - by Gottfried - 02/04/2010, 05:08 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
Question Taylor series of i[x] Xorter 12 10,284 02/20/2018, 09:55 PM
Last Post: Xorter
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 10,503 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Taylor series of cheta Xorter 13 10,967 08/28/2016, 08:52 PM
Last Post: sheldonison
  [rule 30] Is it possible to easily rewrite rule 30 in terms of modular arithmetic ? tommy1729 0 1,453 07/24/2014, 11:09 PM
Last Post: tommy1729
  Generalized arithmetic operator hixidom 16 12,277 06/11/2014, 05:10 PM
Last Post: hixidom
  [integral] How to integrate a fourier series ? tommy1729 1 2,309 05/04/2014, 03:19 PM
Last Post: tommy1729
  Tetration and modular arithmetic. tommy1729 0 1,814 01/12/2014, 05:07 AM
Last Post: tommy1729
  On naturally occuring sums tommy1729 0 1,806 10/24/2013, 12:27 PM
Last Post: tommy1729
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 15,672 02/04/2012, 05:02 AM
Last Post: Kouznetsov
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,672 09/04/2011, 05:59 AM
Last Post: Gottfried



Users browsing this thread: 1 Guest(s)