02/06/2010, 12:52 AM

(02/05/2010, 10:10 PM)Gottfried Wrote: 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.

No, thats not the way it goes. 1+2+4+... = -1 is false in the normal sense of the limit; if it is despite useful in certain circumstances this needs to be mentioned and not the opposite that it is not useful here.

So if you find a certain way to use summation methods in the iteration exponent and getting interesting results - great, but nobody would expect that it works by default.

Quote:(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)

I didnt read such a monograph, but I can not imagine (from what I read by Hardy or Knopp) that they write something that is formally false, like 1+2+4+...=-1.

One term from analysis is for example "absolutely convergent", where you can reorder the summands in any way without affecting the limit. But non-absolutely convergent series is still a convergent series but the limit (of the partial sums) may depend on reordering. Here we not even work with divergent series, and we use no complicated summation method but just reorder the terms - but despite they give different limits. Taking this further you could prove that every number is equal to any other number.