Comments on "From the summation-by-parts algorithm to Pi"

This paper discusses the comment on the issue of the AP-S Magazine by Mosig in December 2005. There, he describes the evaluation of an infinite series by a rearrangement of the terms in the series in a manner that is "...a discrete equivalent of the classical integration-by-parts technique". The desirable performance of this approach is demonstrated by its application to the Leibniz-Gregory series for pi, and found to yield an accuracy of about 10 digits using 50 terms in the original series, compared to the 10⁺¹⁰ terms that would otherwise be required. The summation-by-parts procedure is one of the class of numerical techniques that can be described as convergence-acceleration methods (CAMs). As exemplified by Mosig\´s illustrative example, many ways have been found to evaluate pi, most of which are computationally intensive and not at all practicable unless a convergence-acceleration-method approach is applicable