Tetration/Powertowers - a (zeta) eta-like series. How to determine summability?

In previous postings I proposed results for two types of
alternating series concerning powertowers.

a) of increasing height

AS(b)= 1 -b + b^b - b^b^b + b^b^b^b - ... +

with some numerical results for a range of b even outside the
range for infinite height e^-e < b_classical < e^(1/e) , for instance
b=10; approximate values for such a series, which seem to be not summable
with any current summation-method (see previous postings or [4] or [3]) )

b) of same height, but increasing top-exponent like

GS(b) = b^b^0 - b^b^1 + b^b^2 - b^b^3 +...- ...

and a conjecture, that (somewhat reformulated)

b^b^-inf ... + b^b^-2 - b^b^-1 + b^b^0 - b^b^1 +b^b^2 -...+... =0

for any height of the powertowers (see previous postings or [2])
which embeds the geometric series as special case.

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Here I want to propose a related way to compute the third type of series

c) ET(2,x) = 1^1^x - 2^2^x + 3^3^x + 4^4^x -...+...

where the height of the towers is principally a parameter, but is
set to 2 for a start. (Height 1 gives then the eta-series as special
case).

I did not arrive at certain conjectures about relations concerning
different top-exponents or different heights yet; most of the problem
is due to some unknown behaviour of parts of the computing process.
What I'm missing is a valid description of the rate of divergence for
alternating sums of like powers of logarithms

lambda(m) = log(1)^m - log(2)^m + log(3)^m - ...

for the sequence of m = (0,1,2,3,4,...) and a method to sum the
strongly divergent series ET(h,x) for x>1.(see [1])

Is something known about that function lambda(m) for integer-parameters m,
that I could use here, and can the range of summability of c) be extended
in some way, to check (and possibly extend) my numerical approximations?

Gottfried Helms


[1] http://go.helms-net.de/math/tetdocs/Tetra_Etaseries.pdf


[2] http://go.helms-net.de/math/tetdocs/Tetration_GS_short.pdf


a bit "journalistic"
[3] http://go.helms-net.de/math/tetdocs/10_4_Powertower_article.pdf
more introductory than [3] but less "journalistic"
[4] http://go.helms-net.de/math/tetdocs/10_4_Powertower.pdf

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Gottfried Helms, Kassel



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Gottfried Helms, Kassel

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