Hmm. My hypothesis that the double-sum, or at least the exp-series, can only provide continuum sums of functions for "double-exponential type" or less seems to be wrong. Indeed, it seems that if an exp-series exists and converges for a function, then its continuum sum does too.

Consider the triple-exponential function . This can be expressed as an exp-series

where are the coefficients of the Taylor series of at (MacLaurin series). Taking the continuum sum gives,

As the coefficients of both sums are smaller than those of the original sum (because for all ), if the original series converges at 0 and at the point , so does this (see series comparison test). Since we have an expression for , we have achieved its continuum sum and the proof (or disproof, insofar as my original hypothesis that such a series could not yield a continuum sum of something faster than a double exponential, but as a proof this proves even more, namely that any convergent exp-series' continuum sum also converges, unlike the case with Taylor series summed via direct application of Faulhaber's formula) is complete.

As exp-series look to be a special case of nested series, this suggests even 2 layers of nesting may be able to represent tetration and continuum-sum it, though there's still no proof for that. The special case of exp-series themselves do not appear useful for doing tetration with Ansus' formula, however, for two reasons: any function constructed with them is -periodic, yet tetration seems not to be given pretty much every "good" extension there is -- though they may be able to express tetration for the base whose regular tetration is periodic with the required period (and so the exp-series can be recovered via the Fourier series), but a single base isn't very useful. And, we can't even represent as an exp-series, thus we can't even continuum-sum one exp-series to another exp-series so this is not very useful insofar as trying to iteratively apply Ansus' formula to generate tetrationals goes!

Consider the triple-exponential function . This can be expressed as an exp-series

where are the coefficients of the Taylor series of at (MacLaurin series). Taking the continuum sum gives,

As the coefficients of both sums are smaller than those of the original sum (because for all ), if the original series converges at 0 and at the point , so does this (see series comparison test). Since we have an expression for , we have achieved its continuum sum and the proof (or disproof, insofar as my original hypothesis that such a series could not yield a continuum sum of something faster than a double exponential, but as a proof this proves even more, namely that any convergent exp-series' continuum sum also converges, unlike the case with Taylor series summed via direct application of Faulhaber's formula) is complete.

As exp-series look to be a special case of nested series, this suggests even 2 layers of nesting may be able to represent tetration and continuum-sum it, though there's still no proof for that. The special case of exp-series themselves do not appear useful for doing tetration with Ansus' formula, however, for two reasons: any function constructed with them is -periodic, yet tetration seems not to be given pretty much every "good" extension there is -- though they may be able to express tetration for the base whose regular tetration is periodic with the required period (and so the exp-series can be recovered via the Fourier series), but a single base isn't very useful. And, we can't even represent as an exp-series, thus we can't even continuum-sum one exp-series to another exp-series so this is not very useful insofar as trying to iteratively apply Ansus' formula to generate tetrationals goes!