Arithmetic in the height-parameter (sums, series)

[bo198214]

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(-) .

Of course it is allowed. In this context a series would be considered as a sequence, the sequence of the partial sums; and a good continuous iteration of a function should be continuous in the exponent, i.e.
\( \lim_{n\to\infty} f^{\circ s_n}(x) = f^{\circ \lim_{n\to\infty} s_n}(x) \).

But the partial sums of 1+2+4+8+... as well as the partial sums of 1-1+1-1... diverge.
At most one could say that the partial sums of 1+2+4+8+... converge to oo (e.g. in the topology of the complex sphere convergance to infinity has a well defined meaning.)

So by continuity of the iteration exponent, one would expect
\( f^{\circ 1+2+4+8+\dots} (x) = f^{\circ \infty}(x) \)
and one would expect
\( f^{\circ 1-1+1-1}(x) \) to diverge.
Which is indeed the case.

If you choose the partial sums differently, e.g. by calculating the mean or so then perhaps 1-1+1-1... converges to 1/2 and so this would happen if raised to the iteration exponent (in which case you however would lose the integer exponents).

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.

This is not true. The partial sums of 1-1+1-1... do not converge, the series is divergent. Whether Euler wrote something different in his time is another thing. Calculus wasnt so precise at this time, and in a *certain way* it makes sense to assign the value 1/2, but in the default meaning of what every freshman learns in the first analysis course about series it is provably wrong, false, not true, incorrect, ...

[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")

There is nothing special about taking limits in the iteration exponent, if you take any continuous function h, the same thing occurs:
h(1+2+4+...)=h(oo)
h(1-1+1-1+...) diverges

but sequences of integer numbers never have non-integer numbers as limit.