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 \( b \) greater than \( \eta \) and \( a<b \). But thats exactly the wrong direction. We need to change the base from \( b\le\eta \) to \( a>\eta \). Even if we had a proper change of base formula we need to check that it is consistent for change of bases smaller than \( \eta \), i.e. that it transforms the already known unique solution for base \( b\le\eta \) into the already known unique solution for base \( a\le\eta \).
If we had a change of base formula then we anyway dont need the converging solution for \( b=\eta \) (via \( e^x-1 \)), 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.
