11/13/2009, 12:02 AM
By nth apprimation to intuitive tetration (base 2):
So it seems as though this sequence converges to something other than zero.
If you're looking for a base with "nice" properties, I would recommend base (3.0885325).
What makes base (3.0885325) special is that it is the maximizer of \( ([4]x)^{-1}(x) \), and maximum of \( x[5]\infty \) over the interval (?, 1.6353245). I think this is analogous to how base (e) is the maximizer of \( ([3]x)^{-1}(x)=\sqrt[x]{x} \) and the maximum of \( x[4]\infty \) over the interval (0.065988, 1.44467).
Code:
n sexp_2'(0) = -slog_2''(1)/slog_2'(1)^3
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5 0.0108519
10 0.0171036
15 0.0173285
20 0.0173507
25 0.0173537
30 0.0173539So it seems as though this sequence converges to something other than zero.
If you're looking for a base with "nice" properties, I would recommend base (3.0885325).
What makes base (3.0885325) special is that it is the maximizer of \( ([4]x)^{-1}(x) \), and maximum of \( x[5]\infty \) over the interval (?, 1.6353245). I think this is analogous to how base (e) is the maximizer of \( ([3]x)^{-1}(x)=\sqrt[x]{x} \) and the maximum of \( x[4]\infty \) over the interval (0.065988, 1.44467).