jaydfox Wrote:Essentially, if k is the number of terms at which we truncate the series expansion, then there is a non-zero radius for which the series is initially convergent (i.e., the root-test for terms 1 through k would all be less than 1).

I dont understand a word. If you truncate the series, it is a polynomial and a polynomial is defined on the whole complex numbers, i.e. infinite radius of convergence.

Quote:Regardless, if the proof has already been shown, then combined with my change of base formula, we now have a unique solution to tetration of bases greater than eta.As it appears to me your change of base formula works merely for base greater than and . But thats exactly the wrong direction. We need to change the base from to . Even if we had a proper change of base formula we need to check that it is consistent for change of bases smaller than , i.e. that it transforms the already known unique solution for base into the already known unique solution for base .

If we had a change of base formula then we anyway dont need the converging solution for (via ), because we could simply compute that solution from a smaller base.

Quote:By the way, for the reference to Ecalle, where can I get a copy, and more importantly, is there an English translation available?Not sure, I lent it from the library. There seems no translation into english.