(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