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Arithmetic-Geometric Means Revisited

Borwein, Jonathan M. and Lisonek, Petr and Macdonald, John A. (1997) Arithmetic-Geometric Means Revisited. MapleTech, Special Issue on Maple in the Mathematical Sciences, 4 . pp. 20-27.

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      Abstract

      For any integer $N\ge 2$ we study a two-term recurrence (iteration) $AG_N$ consisting of two sequences which converge rapidly ($N$th-order) to a common limit. Our goal is to identify this limit. Using computer algebra we are able to independently discover and prove some very classical results, e.g.~for $N=2$, in an automated way. We also present a new, self-contained proof of the limit formula for $N=3$. Another objective of this paper is to open the way of investigating the cases $N>3$ using computer algebra. The definitions and results are collected in the first half of the paper. In the second half we then present Maple code for proving certain claims made in the first part. We also present illustrations of using the iterations $AG_N$ for high-precision evaluations of special functions.

      Item Type: Article
      Additional Information: pubdom FALSE
      Subjects: 33-xx Special functions
      40-xx Sequences, series, summability
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 27 Oct 2003
      Last Modified: 14 Sep 2014 22:01
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/4

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