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 Tetra-series Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 11/20/2007, 12:47 PM (This post was last modified: 11/20/2007, 03:21 PM by Gottfried.) 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)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) = xor, 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 $\hspace{24} 0 = \sum_{h=-\infty}^{+\infty} T_b(x,h)$ 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 « Next Oldest | Next Newest »

 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

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