Change of base formula for Tetration

[jaydfox]

There was also no proof for the convergence of your change of base formula. Though it looks for me that you could make it.
For the convergence of \( e^z-1 \) I have really strange news, see the corresponding thread (later).

Really, I thought it obvious it converges. As n goes to infinity, the epsilon goes to 0 (and quite rapidly at that), making the formula exact in the limiting case. Because the limiting case is exact and the formula converges very fast, the solution is even practical.

The proof relies on bases greater than eta, but if one starts with one of the bases greater than eta (e seems the obvious choice), and uses iterated logarithms for the other base (see Andrew Robbins's notation), then one can even get solutions for bases between 1 and eta, though I'm not sure if there is a defensible "unique" definition of the zeroeth iterate. (more on that when I get to bases less than eta).