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+--- Thread: Series of tetrates:consecutive bases (/showthread.php?tid=368)
Series of tetrates:consecutive bases - Gottfried - 10/20/2009
Just came across an article, which refers to the alternating series
\( \hspace{48} s= - \sum_{b=2}^{\infty} (-1)^b *log( (\frac1b)\^\^ ^{\tiny 2}) \)
whose terms contain the towers of height 2 of reciprocals of consecutive bases.
The terms of the series in their original notation are
\( \hspace{48} (-1)^b * \frac{\log(b)}{b} \)
which I converted to
\( \hspace{48} = \log(b^{\frac1b}) = - log(\frac1b ^{\frac1b})= - log(\frac1b\^\^ ^{\tiny 2}) \)
The result is -without explicite derivation, but the method was indicated- given as
\( \hspace{48} s= \log(2)(\gamma - \frac12\log(2)) \)
Article: Convergence acceleration of series
Pascal Sebah and Xavier Gourdon
http://numbers.computation.free.fr/Constants/Miscellaneous/seriesacceleration.ps
January 10, 2002
Note: a couple of msgs of mine related to such series you'll find by keyword "tetra-eta-series", which I used earlier