Arithmetic in the height-parameter (sums, series)

[Gottfried]

(*1) I'm proudly linking to my own solution for the summation of 0!-1!+2!-3!... in eulerian-matrix chap 3.2

Seems matrix method is quite potent to sum everthing

- unfortunately not... That was subject to a very recent post to sci.math, where I asked, whether there is a known limit for the summation-power of triangular matrices for matrix-based transform & summation. I didn't find a possibility for a more powerful matrix and suppose, that such a limit might already be known.

Well, back to the main subject. Surely, the series 1+2+4+8+... is a "fragile" or let's say, basing arguments on it is fragile. I chose that only because of its simple occurence. Generally I think, it should be allowed to formulate the iteration parameter as a series, and also as a powerseries in x, for instance

\( f(x) = \exp_b^{^o ^{1+x+x^2+x^3+...}} (a) \)

and then discuss the limiting behaviour, when abs(x)->1(-) . One of the reasons, that the summation 1-1+1-1+... made it into serious math, and was accepted to be identified with the value 1/2 was the discussion of the form of g(x) = 1+x+x^2+... for x=-1, the form of the geometric series, its translation into the closed form 1/(1-x) and finally the method of analytical continuation.
But can we identify the above f(-1) with \( f(-1) = \exp_b^{^o ^{0.5}}(a) \) using the rationale of evaluation using the partial sums?
Would be interesting, whether this makes sense, anyway ...

So do we deny the validity of divergent summation in the height-parameter completely? Then we should do it also explicitely, for instance also in a remark in wikipedia or other online-resources.
But I think, that were a step too early.
[Update] a) What would we do in cases, where the height-parameter is expressed as zeta-series. Zeta-regularization is a well established procedere. Does it produce contradictions if inserted in the height-parameter in tetration (or other iterated functions) ?[/update]
b) We should in general look, whether there are possibilities, where divergence/summation keeps a sensical result, or if the contrary occurs, and we cannot find any such meaningful result, then we should try to explain, why the consideration near the limit can*not* be extended beyond (or some wording like "analytical continuation makes no sense here")

Hmm, perhaps I'd look for opinions in sci.math, too...

Gottfried