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The enhancement of Machin’s formula by Todd’s process

Published online by Cambridge University Press:  01 August 2016

Michael Wetherfield
Michael Wetherfield*
Affiliation:
8 Bafford Lane, Charlton Kings, Cheltenham GL53 8DL
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Extract

1. Background

From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:

which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is

Writing x = cotθ, the series becomes

Expressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:

Type
Articles
Information
The Mathematical Gazette , Volume 80 , Issue 488 , July 1996 , pp. 333 - 344
Copyright
Copyright © The Mathematical Association 1996

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References

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