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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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The arithmetic-harmonic mean
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by D. M. E. Foster and G. M. Phillips PDF
Math. Comp. 42 (1984), 183-191 Request permission

Abstract:

Consider two sequences generated by \[ {a_{n + 1}} = M({a_n},{b_n}),\quad {b_{n + 1}} = M’ ({a_{n + 1}},{b_n}),\] 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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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Math. Comp. 42 (1984), 183-191
  • MSC: Primary 40A99; Secondary 40A25
  • DOI: https://doi.org/10.1090/S0025-5718-1984-0725993-3
  • MathSciNet review: 725993