This also makes me think about the nature of the continuum sum operator as a subject in itself. Examining the formula for the continuum sum of a function that converges quickly to a limit at infinity, it is important to note its extremely "global" nature: unlike say, the derivative, which depends on the value of the function only very near ("infinitely" near) the point at which it's taken, or even the integral, which depends on the values over some interval, but which may be finite and bounded, the continuum sum at a point not only depends on the value there, but on that at the point plus 1, at the point plus 2, at the point plus 3, and so on, all the way to infinity, and on the value at the point 1, at the point 2, at the point 3, and so on, to infinity as well! (if we start our summation at index 0) In other words, almost any change of any substantial nature, even very far away along the number line or right complex half-plane, that causes little or no change in the function near the origin (or whatever the lower bound on the summation is) -- doesn't matter if its 10 numbers away or 10 billion or a googolplex or a giggolplex (i.e. \( ^{^{100} 10} 10 \)) or a moser or even Graham's Number of places away -- can still cause significant change in the continuum sum there.
This might be part of the reason "why" Faulhaber's formula fails to converge (beyond the obvious one that its terms don't settle: why don't they, on a deeper level?) for series with finite convergence radii: although locally they look much like the function they are approximating, globally they don't at all: the truncated series take on huge values over most of the real line or the complex plane. If the general "canonical" continuum sum operator (which, I believe, is what we need to construct the general tetration of arbitrary complex bases to arbitrary complex heights or towers: Ansus' formula bridges the tetration problem to the continuous summation problem -- if we have a "natural" definition for continuum sum, then it only makes sense tetration should be consistent with it through the Ansus formula) is also "global" like this, and the Faulhaber formula is a special case of that general operator, which makes sense (it can actually be obtained fairly naturally from the convergent continuum sum special case I mentioned, I can show you how it's done if you want and don't know already), then it should not be surprising that it will not work.
This might be part of the reason "why" Faulhaber's formula fails to converge (beyond the obvious one that its terms don't settle: why don't they, on a deeper level?) for series with finite convergence radii: although locally they look much like the function they are approximating, globally they don't at all: the truncated series take on huge values over most of the real line or the complex plane. If the general "canonical" continuum sum operator (which, I believe, is what we need to construct the general tetration of arbitrary complex bases to arbitrary complex heights or towers: Ansus' formula bridges the tetration problem to the continuous summation problem -- if we have a "natural" definition for continuum sum, then it only makes sense tetration should be consistent with it through the Ansus formula) is also "global" like this, and the Faulhaber formula is a special case of that general operator, which makes sense (it can actually be obtained fairly naturally from the convergent continuum sum special case I mentioned, I can show you how it's done if you want and don't know already), then it should not be surprising that it will not work.
