Just came across an article, which refers to the alternating series

whose terms contain the towers of height 2 of reciprocals of consecutive bases.

The terms of the series in their original notation are

which I converted to

The result is -without explicite derivation, but the method was indicated- given as

Article: Convergence acceleration of series

Pascal Sebah and Xavier Gourdon

http://numbers.computation.free.fr/Const...eration.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

whose terms contain the towers of height 2 of reciprocals of consecutive bases.

The terms of the series in their original notation are

which I converted to

The result is -without explicite derivation, but the method was indicated- given as

Article: Convergence acceleration of series

Pascal Sebah and Xavier Gourdon

http://numbers.computation.free.fr/Const...eration.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

Gottfried Helms, Kassel