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

DOI:
https://doi.org/10.1090/S0025-5718-1984-0725993-3

MathSciNet review:
725993

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider two sequences generated by

*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 . 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.

**[1]**B. C. Carlson, "Algorithms involving arithmetic and geometric means,"*Amer. Math. Monthly*, v. 78, 1971, pp. 496-505. MR**0283246 (44:479)****[2]**B. C. Carlson, "An algorithm for computing logarithms and arctangents,"*Math. Comp.*, v. 26, 1972, pp. 543-549. MR**0307438 (46:6558)****[3]**D. M. E. Foster & G. M. Phillips, "A generalization of the Archimedean double sequence,"*J. Math. Anal. Appl.*(In press.) MR**748590 (85g:40001)****[4]**T. L. Heath,*A History of Greek Mathematics*, Vol. 2, Oxford, 1921.**[5]**George Miel, "Of calculations past and present: The Archimedean algorithm,"*Amer. Math. Monthly*, v. 90, 1983, pp. 17-35. MR**691010 (85a:01006)****[6]**G. M. Phillips, "Archimedes the numerical analyst,"*Amer. Math. Monthly*, v. 88, 1981, pp. 165-169. MR**619562 (83e:01005)****[7]**E. T. Whittaker & G. N. Watson,*A Course of Modern Analysis*, 4th ed. Cambridge Univ. Press, Cambridge. 1958. MR**1424469 (97k:01072)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0725993-3

Keywords:
Arithmetic-harmonic mean,
Archimedean process

Article copyright:
© Copyright 1984
American Mathematical Society