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Mathematics of Computation

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The arithmetic-harmonic mean

Authors: D. M. E. Foster and G. M. Phillips
Journal: Math. Comp. 42 (1984), 183-191
MSC: Primary 40A99; Secondary 40A25
MathSciNet review: 725993
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Abstract: Consider two sequences generated by

$\displaystyle {a_{n + 1}} = M({a_n},{b_n}),\quad {b_{n + 1}} = M' ({a_{n + 1}},{b_n}),$

where the $ {a_n}$ and $ {b_n}$ are positive and M and M' are means. The paper discusses the nine processes which arise by restricting the choice of M and M' to the arithmetic, geometric and harmonic means, one case being that used by Archimedes to estimate $ \pi $. Most of the paper is devoted to the arithmetic-harmonic mean, whose limit is expressed as an infinite product and as an infinite series in two ways.

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Keywords: Arithmetic-harmonic mean, Archimedean process
Article copyright: © Copyright 1984 American Mathematical Society