Road testing Ansus' continuum product formula

[mike3]

Interestingly, though, it seems Borel summation might work with the Faulhaber (and probably also this) formula, due to the not-too-bad divergence of the terms. I tried sum numerical tests using 1 integral and 60 derivatives, but that only seemed to get the Borel-function (the thing inside the Borel-summation integral) to converge in a radius of around 13, which isn't much, yet integrating up to that limit (from 0 to 13) got the coefficient of x^2 in the Tayor expansion of the sqrt(2) regular tetration's continuum sum at 0 as 0.439, and the coefficient of x as 0.666, both of which seem to agree with differentiation of the left hand side of the sum formula (the log with the quotient of the derivative of tetration, etc. inside it bit), though as I mentioned, I can't try for more accuracy than this (as I'll need to have to actually try generating series expansions of tetration for other bases, esp. the fabled b = 0.04, which I think the Borel summation may be more suited to as it doesn't require all those wacky parameters the other did) due to limited convergence radius. Yet the theory mentions about analytically continuing. There's also the possibility of trying a higher-order one (adds another factorial in the denominator) with double integrals, yet that would be even nastier to compute due to the whole 2d grid thing.

However, I'm not sure how you'd analytically continue the sum in the Borel integrals past its convergence radius on the positive real axis to get more accuracy.

Is there any good code for doing Borel summation that I could test with?