02/27/2013, 06:40 PM
(This post was last modified: 02/28/2013, 11:32 AM by sheldonison.)

(02/27/2013, 02:19 PM)Balarka Sen Wrote: I made an observation : for very small values of z, it seems likely that as b tends towards infinity, b^^z grows to infinity too, but rather slowly. I mean for allIts not a trivial question. And I had a difficult time getting my kneser.gp algorithm to converge for large bases, though it currently works for b>100000. Here is a related question, that may lead to a fruitful investigation path, and possibly a proof. Can you prove that the for arbitrarily large base=b approaches arbitrarily close to zero? There is a fairly simple linear approximation one can use for tetration for arbitrary bases, that is continuous, and has a continuous first derivative, and works surprisingly well. The estimation uses a straight line linear estimate between and . For example, for base e, the linear approximation is sexp(-1)=0, sexp(1)=1, with a straight line in between, and .

edit: updated approximation equations, and plot

If I did my algebra correctly, than using the linear approximation for sexp for arbitrary bases leads to the estimate for bases>=e and the region from k-1..k has an exponential approximation of . For z>k the approximation switches over to a double exponent. This exponential approximation assumes the linear region includes sexp(0), which is true if base>=e.

This approximation gives sexp(0)=1, and sexp(1)=b, which are both exact. Then you you could conjecture that for large enough bases (empirically, b>9), the actual sexp(k)>e. Also, I would conjecture that for b>9 the approximation is less than actual sexp(z) until z=1, where by definition the approximation is exactly correct once again.

Anyway, such an slog(e) approximation for large bases goes to zero, but very slowly. For b=googleplex=, the approximation is slog(e)=0.0043. For b=10, the approximation is 0.545 and the correct . For b=100, the approximation is 0.396 and the correct value is . For b=100000, the approximation is 0.290, and the correct value is . Here is a graph of sexp_100000(z). The function is surprisingly well behaved in the region of interest. Here, , and the linear approximation region would be from -1.71 to -0.71. The actual sexp is in red, and the linear approximation is in green.

- Sheldon