Growth of superexponential

[sheldonison]

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 \( \lim_{b \rightarrow \infty} {}^{z} b \rightarrow \infty \) for all \( z > 0 \)

Its 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 \( \text{slog}_b(e) \) 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 \( \log_b(\log_b(e)) \) and \( \log_b(e) \). For example, for base e, the linear approximation is sexp(-1)=0, sexp(1)=1, with a straight line in between, and \( \text{slog}_e(e)=1 \).

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 \( \text{slog}_b(e)\approx \frac{1}{1+\log(\log(b))}=k \) and the region from k-1..k has an exponential approximation of \( \text{sexp}_b(z)\approx\exp(\frac{z}{k}) \). 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=\( 10^{10^{100}} \), the approximation is slog(e)=0.0043. For b=10, the approximation is 0.545 and the correct \( \text{slog}_{10}(e) \approx0.544 \). For b=100, the approximation is 0.396 and the correct value is \( \text{slog}_{100}(e) \approx0.374 \). For b=100000, the approximation is 0.290, and the correct value is \( \text{slog}_{100000}(e) \approx0.250 \). Here is a graph of sexp_100000(z). The function is surprisingly well behaved in the region of interest. Here, \( k\approx 0.29 \), 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