12/12/2009, 09:53 PM

(12/12/2009, 08:18 PM)mike3 Wrote: *sighs* I guess its back the drawing board, then. This continuum sum thing seems really difficult to generalize to arbitrary analytic functions, as is needed for tetration.

Well, even if there was a simple Mittag-Leffler expansion, the chance to have luck with a convergent Faulhaber application, was anyway quite low, wasnt it? If not even entire the continuum sum of exp(exp(x)) was convergent.

Hm, so summarizing:

exp(x) has a convergent Faulhaber sum.

non-entire functions have no convergent Faulhaber sum.

functions with over-exponential growth (like exp(exp(x))) have probably no convergent Faulhaber sum.

However if there was a way to rearrange the terms, there might convergence be possible (as you greatly showed with exp(exp(x)) and 1/(x+1)).

But then my question would be whether the value depends on how the terms are rearranged (I mean not only giving different branches, but a continuum of solutions).