Devilish Tricks for Sequence Acceleration

Abstract

The two most famous quotes about divergent series are Abel's "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever," and Heaviside's "This series is divergent, therefore we may be able to do something with it." Today a lot more is known about divergent series than in either's day, so we can say now that, on balance, Heaviside wins, and we now have plenty of license to use divergent series. This article talks about some "well-known" methods (that is, well-known to experts) to do so, and in particular talks about some of the devilishly good features of evalf/Sum, long one of my favourite tools in Maple. But Abel had a point, too, and we'll see some "shameful" things, which will give the reader some necessary caution to go along with their license.

Special Functions in Problem Solving Environments: A personal view

“…See for instance[12] for a discussion of why this particular average is the right one for combining an estimate from the midpoint rule with an estimate from the trapezoidal rule. In short, it's because the midpoint rule is usually twice as accurate as the trapezoidal rule, and its error is usually the opposite sign.…”