(11/27/2012, 05:13 PM)Nasser Wrote: [ -> ]solve this limit ....I don't know if there is a software support tetration!!

I posted a pari-gip routine that generates sexp(z) for real bases greater than

here,

http://math.eretrandre.org/tetrationforu...hp?tid=486.

By definition, T(a,0) = 1, since sexp(0) is defined to be 1. If T is analytic, then for each value of a, T has a Taylor series expansion around 0, corresponding to the Taylor series for sexp(z) around 0. Define

as the first derivitive of that Taylor series.

There is an unproven conjecture that

is analytic in the base=a for complex values of a, with a singularity at base

. For real values of a, if

, then sexp(z) goes to infinity at the real axis as z increases. If

, then iterating

converges towards the attracting fixed point as n goes to infinity, but this is a different function than tetration. Then for base>

, we can have a taylor series for the any of the derivatives of

, with the radius of convergence =

.

I posted such a the taylor series for the first derivative of the base. For base=e, the first derivative ~= 1.0917673512583209918013845500272. The post includes pari-gp code to calculate sexp(z) for complex bases; the code for complex bases isn't as stable as the code for real bases, and doesn't always converge. If you're interested in a Taylor series for

for your limit, search for "the Taylor series of the first derivative of sexp_b(z), developed around b=2" in this post:

http://math.eretrandre.org/tetrationforu...e=threaded.

- Sheldon