Integration?

[Gottfried]

[quote='73939' pid='4939' dateline='1278102954']
(...)
This time a bit more careful with the last digits I would say: \( \text{Ti}_3(1)\approx 0.573121 \).

Pari/GP gives using numerical integration and internal float-precision of 200 digits

gp> intnum(x=-1,0,(1+x)^(1+x)^(1+x))
%1 = 0.573121567043619582758830

[update] I also had tried it with my matrix-formulation and my simple possibilities of convergence-acceleration using Euler- et al summation.
Having the same taylor-series with which Henryk worked, an improvement would only be possible, if such a convergence-acceleration were effective. But what I see if I look at the terms of the powerseries of the integral with the powers of x(=0 and =-1) mutiplied in, is that the summands decrease slowly to zero (?) but not with alternating sign - so common convergence-accelerators should not work easily. (The extreme case would be to try to sum the harmonic series, where such convergence acceleration would even be contradictory).
So with all my usual tools I come not nearer than four to six digits (using 64 terms) like Henryk, - thus, to apply this model of powerseries at all, would require some more consideration before ...
[/update]
Gottfried