Arithmetic in the height-parameter (sums, series)

[Gottfried]

Well, the equality 1 + 2 + 4 + 8 + ... = -1, is very fragile if you want to take it as a serious argument. It doesnt surprise me that you dont get the result you expect.

Well ... "fragility"...., a bit informal.

What I'm trying is to improve my general understanding of infinite series and especially that of divergent series (well in the context of iteration and tetration). Perhaps I went too far, in that I (unconsciously) moved nearer and nearer to a notion of "whenever a sum for a divergent series can be established (for instance by cesaro-sum), we can use that value in the instance of the series".
That's obviously a false mental notion - although it is suggestive and is working in many cases. (*1)

Perhaps a step towards a better understanding is the following consideration.

Assume the sequence of partial sums of some divergent series a0+a1+a2+... or even better, using some limit notation like
f(x) = a0+a1 x + a2 x^2 + ... for x->1(-)
Then for some divergent series we take the mean of the sequence of partial sums, if that converges.

But in my two examples we deal (with the limit of) functions of the partial sums and the condition "...if that converges..." must be extended to the additional condition "...if the sequence of function-evaluations of the partial sums converge..." or something like. (*2)

Hmm. I think to proceed here I should do some examples, where the function-evaluations converge and some, where they don't ...

Gottfried


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

[update]
(*2) well, again in a second thought: I was actually thinking about that, when I posted this whole problem: we see, that the tretration-function even provides this convergence of sequence of function-evaluations. And still this is not enough to guarantee the usability of the "value" of the divergent series...
[/update]