I really have to put my foot down on this one. The lower limit is is not zero as it appears in the Tetration_Summary page. It is 1. I have re-derived a more general formula for this that accentuates this lower index:

as you can see from this, if the final derivative in the denominator is evaluated at (

), then this means

, which means the lower index of the product is (k=1), not (k=0).

@Ansus

Your derivations are based on the (k=0) formula (which is wrong), but other than that, they are

quite clever! I never thought to do that. I think there would be less room for error if we use the "P" function to simplify things. Starting with the basic derivatives:

combining them gives:

which is about as rigorous as I can make it, so that should be right.

Andrew Robbins