Arithmetic in the height-parameter (sums, series)

[Gottfried]

Hmm, some points come out clearer now.
What I want is not sort of legimitation for this or that but of understanding. I'm ok, if it comes out that we must dismiss any sort of divergent summation in the iteration parameter of functions. What I want is: is this necessary? Are there exceptions?
If this is a general property, I also want, that we add some remark to our interpretation of tetration: "no divergent series in iteration parameter" - be it in open-text collections like wikipedia or in journals.
Since it seems, that is a more general property, I'd like to see this also in articles about summation of divergent series: "that concept is limited to <...something...>" and "not meaningful for <...example:iteration of functions...> " (I need not Euler to assign a fairly general availability of divergent summation - K.Knopp and G.H.Hardy have even dedicated monographies (or monographic-like chapters) to that concept - without mentioning circumstances, where it is *generally* not applicable)

But well, let's see. I think I'll do some more examples first, to improve my own understanding. I'll reply to this all later again.

Gottfried