Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Tetra-series
#1
I have another strange, but again very basic result for the alternating series of powertowers of increasing height (I call it Tetra-series, see also my first conjecture at alternating tetra-series )

Assume a base "b", and then the alternating series

Code:
.    
  Sb(x) = x - b^x + b^b^x - b^b^b^x +... - ...

and for a single term, with h for the integer height (which may also be negative)
Code:
.
  Tb(x,h) = b^b^b^...^x     \\ b occurs h-times

which -if h is negative- actually means (where lb(x) = log(x)/log(b) )
Code:
.
  Tb(x,-h) = lb(lb(...(lb(x))...)   \\ lb occurs h-times

-------------------------------------------------------

My first result was, that these series have "small" values and can be summed even if b>e^(1/e) (which is not possible with conventional summation methods). For the usual convergent case e^(-e)<b<e^(1/e) the results can be checked by Euler-summation and they agree perfectly with the results obtained by my matrix-method.(see image below)


Code:
matrix-notation
Sb(x) = (V(x)~ * (I - Bb + Bb^2 - Bb^3 + ... - ...)  )[,1]
       = (V(x)~ * (I + Bb)^-1 )   [,1]
       =  V(x)~ * Mb[,1]                \\ (at least) for all b>1
       = sum r=0..inf  x^r * mb[r,1]  

serial notation
       = sum h=0..inf  (-1)^h* Tb(x,h)  \\ only possible for e^(-e) < b < e^(1/e)
                                        \\ Euler-summation required


-------------------------------------------------------


Now if I extend the series Sb(x) to the left, using lb(x) = log(x)/log(b) for log(x) to base b, then define

Code:
.
   Rb(x) = x - lb(x) + lb(lb(x) - lb(lb(lb(x))) +... - ...

This may be computed by the analoguous formula above to that for Mb from the inverse of Bb:
Code:
.
   Lb = (I + Bb^-1)^-1

I get for the sum of both by my matrix-method
Code:
.
  Sb(x) + Rb(x) = V(x)~ *Mb[,1] + V(x)~ * Lb[,1]
                = V(x)~ * (Mb + Lb)[,1]
                = V(x)~ *    I [,1]
                = V(x)~ *   [0,1,0,0,...]~  
                = x

  Sb(x) + Rb(x) = x
or, and this looks even more strange (but even more basic)

Code:
.
  0 = ... lb(lb(x)) - lb(x) + x - b^x + b^b^x - ... + ...

x cannot assume the value 1, 0 or any integral height of the powertower b^b^b... since at a certain position we have then a term lb(0), which introduces a singularity.


Using the Tb()-notation for shortness, then the result is



and is a very interesting one for any tetration-dedicated...

Gottfried
-------------------------------------------------------

An older plot; I used AS(s) with x=1,s=b for Sb(x) there.
(a bigger one AS

   
Gottfried Helms, Kassel
Reply


Messages In This Thread
Tetra-series - by Gottfried - 11/20/2007, 12:47 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM
RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM
RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM
RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM
RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM
RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM
RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM
RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM
RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM
RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM
RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM
Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM
RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM
RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM
RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
Question Taylor series of i[x] Xorter 12 10,278 02/20/2018, 09:55 PM
Last Post: Xorter
  Taylor series of cheta Xorter 13 10,949 08/28/2016, 08:52 PM
Last Post: sheldonison
  Derivative of E tetra x Forehead 7 8,657 12/25/2015, 03:59 AM
Last Post: andydude
  [integral] How to integrate a fourier series ? tommy1729 1 2,301 05/04/2014, 03:19 PM
Last Post: tommy1729
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 15,658 02/04/2012, 05:02 AM
Last Post: Kouznetsov
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,670 09/04/2011, 05:59 AM
Last Post: Gottfried
  What is the convergence radius of this power series? JmsNxn 9 14,850 07/04/2011, 09:08 PM
Last Post: JmsNxn
  An alternate power series representation for ln(x) JmsNxn 7 12,915 05/09/2011, 01:02 AM
Last Post: JmsNxn
  weird series expansion tommy1729 2 4,020 07/05/2010, 07:59 PM
Last Post: tommy1729
  Something interesting about Taylor series Ztolk 3 6,069 06/29/2010, 06:32 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)