Vol. 5, No. 4, 2012

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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

${\left\{\frac{{t}^{12m}{\left(t-\left(2-\sqrt{3}\right)\right)}^{12m}}{1+{t}^{2}}\right\}}_{m\in ℕ}$

we produce a family of efficient polynomial approximations to arctangent on the interval $\left[0,2-\sqrt{3}\right]$, and hence provide approximations to $\pi$ via the identity $arctan\left(2-\sqrt{3}\right)=\pi ∕12$. We turn the approximations of $\pi$ 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 $\pi$
Primary: 41A10
Secondary: 26D05