(12/14/2012, 07:16 PM)Gottfried Wrote: Here is some explanation in terms of Pari/GP-code, how the serial alternating iteration sums asum(x) can be expressed/computed with the help of power series.Just a short, but useful addendum: if we assume identity of the serial comnputation of asum(x) via the Pari/GP-sumalt-procedure and that via the Neumann-series-matrices asum_mat(x) and its power series then one should consider to use that second method as its standard basis. I toyed a bit around with differentating and integrating using the asum(x) and found, that the asum_mat(x) needs only about 1/20 of the computation time, so my example integral needed 80 000 msec with the serial implementation of the asum(x) but only 4 000 msec using the power-series implementation.

This should also be useful for the computation of the inverse of asum(x) as long as we need to interpolate it by binary search/Newton-method, where many function calls are needed.

I think moreover, that this shall prove useful, once we shall step further to analytically continue the range for the x and for the base b outside the "safe intervals" and enter the realms of truly divergent series for the asum(x).

Gottfried

Additional readings:

An early(2008 ) discussion of this method and some of the problems, which we seemingly can resolve now, but also a (very natural) view into regions of bases outside the Euler-summable range for the serial computation of the asum(x) is here http://go.helms-net.de/math/tetdocs/Tetr...roblem.pdf

An involved discussion (2007) about the ability of the Neumann-type matrix for the asum(x) to represent an analytical continuation for the divergent cases - the matrix-ansatz was crosschecked against a shanks-summation in the range, where the shanks-summation was computable: http://go.helms-net.de/math/tetdocs/Iter...tion_1.htm

Gottfried Helms, Kassel