Vol. 5, No. 4, 2012

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ISSN: 1944-4184 (e-only)
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A new series for $\pi$ via polynomial approximations to arctangent

Colleen M. Bouey, Herbert A. Medina and Erika Meza

Vol. 5 (2012), No. 4, 421–430
Abstract

Using rational functions of the form

{t12m(t (2 3))12m 1 + t2 }m

we produce a family of efficient polynomial approximations to arctangent on the interval [0,2 3], and hence provide approximations to π via the identity arctan(2 3) = π12. We turn the approximations of π into a series that gives about 21 more decimal digits of accuracy with each successive term.

Keywords
polynomial approximations to arctangent, approximations of $\pi$, series for $\pi$
Mathematical Subject Classification 2010
Primary: 41A10
Secondary: 26D05
Milestones
Received: 25 August 2011
Revised: 30 January 2012
Accepted: 4 March 2012
Published: 14 June 2013

Communicated by Kenneth S. Berenhaut
Authors
Colleen M. Bouey
Mathematics Department
Loyola Marymount University
1 LMU Drive
Los Angeles, CA 90045
United States
Herbert A. Medina
Mathematics Department
Loyola Marymount University
1 LMU Drive
Los Angeles, CA 90045
United States
Erika Meza
Mathematics Department
Loyola Marymount University
1 LMU Drive
Los Angeles, CA 90045
United States