02/20/2009, 02:51 PM
(This post was last modified: 02/20/2009, 05:15 PM by sheldonison.)

bo198214 Wrote:But this question is different from whether converges to a constant, if we assume . If does converge for then - as goes to infinity - also converges. But this means that can not converge to a constant and vice versa.

typo: is what converges to a cyclic 1-cycle function as x grows larger and may converge to a constant as x grows larger for some definitions of slog/sexp.

Quote:But it never gets completely linear, doesnt it? Otherwise it would not be analytic.no, of course not. But I think the contributions of the higher order terms versus a linear estimate between the second and third terms, or between and , becomes insignificant. I plan to try and show that the second and higher order derivatives for the two points, contribute an insignificant delta as b approaches , and that in the limit, the linear term dominates the higher order terms by an arbitrarily large amount, and that the linear approximation suffices as a definition for the tetration for base b.

I had originally intended this thread to be a search for other links about about base changes for tetration. I seem to have gotten side tracked on to defining the sexp function for real numbers for bases .

- Sheldon Levenstein