A method for an infinite sum of finite powertowers

To place a known keyword in the subject line
and to avoid possible confusion I rephrased the
subject line of the older thread to:

A method for an infinite sum of finite powertowers

Description of the method:
http://go.helms-net.de/math/binomial_new/10_4_Powertower.pdf

Note the difference between this and the analysis
of the "infinite powertower" itself, which is
not directly involved, (though the analytical bounds
for its base-parameter play an important role, anyway)


It provides a method for a somehow analogy to
the shorthand formula for geometric series,
where only finite elements (powertowers)
of ascending height are (alternating) summed.
Similarly, as the evaluation of the geometric series
(infinite sum of ascending finite powers) can be
analytically continued for the classical divergent
cases, this method seems to perform such a continuation
for the infinite sum of finite powertowers for
some divergent cases.


Base concepts of summation of divergent series
are involved. For convergent cases the method
provides the same solutions as conventional
infinite summation of alternating series
(Cesaro or Euler-summation). The range for the
permitted domain is extended beyond the critical
upper bound for the convergence of the
*infinite powertower*.

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

Two detailed dokumentations of results in the
critical regions of the parameters s

* checking e^-e < s <7 in small steps
http://go.helms-net.de/math/binomial_new/powertower/powertowertables.htm

* check of an additional, methodspecific singularity at e^(-1)

http://go.helms-net.de/math/binomial_new/PowertowerproblemDocSummation.htm

Comments welcome, since the estimations for
convergence-/summability criteria need be
enhanced.

Gottfried

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Gottfried Helms, Kassel
.