Tetra-series

[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

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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

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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

\( \hspace{24}
0 = \sum_{h=-\infty}^{+\infty} T_b(x,h)
\)

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

Gottfried

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