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An infinite product of nested radicals for log x from the Archimedean algorithm

Published online by Cambridge University Press:  14 June 2016

Thomas J. Osler,
Walter Jacob and
Ryo Nishimura
Thomas J. Osler
Affiliation:
Mathematics Department, Rowan University, Glassboro NJ 08028USA e-mail: osler@rowan.edu
Walter Jacob
Affiliation:
Mathematics Department, Temple University, Philadelphia, Pa. USA
Ryo Nishimura
Affiliation:
Department of Frontier Materials, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya Aichi, 466-8555, Japan

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The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations

(1)

and

(2)

(Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a0, the smaller has perimeter b0. (Archimedes used hexagons with and b0 = 3, but regular polygons of any number of sides can begin the iterations.) Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated, from (2) and (3). (See [1] for a derivation or many explanations on the web.) Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon.

Information

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 548 , July 2016, pp. 274 - 278
Copyright
Copyright © Mathematical Association 2016 

References

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