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Something interesting about Taylor series
#4
(06/24/2010, 08:23 PM)tommy1729 Wrote: http://mathworld.wolfram.com/PowerTower.html

formula ( 6 ) , ( 7 ) and ( 8 ).

a classic.

Hm, that is a development in , Not really a Taylor devlopment of x^^n at some point .
I wonder whether we have formulas for the powerseries development of at (and not at 0 because the powertower is not analytic there).

And indeed Andrew pointed it out in his tetration-reference formula (4.17-4.19) (or in the Andrew's older Tetration FAQ 20080112: (4.23-25)):


where:

otherwise:


I think it has convergence radius 1 because the substituted logarithm has convergence radius 1 and also because of the singularity at 0.
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