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A new series for \(\pi\) via polynomial approximations to arctangent. (English) Zbl 1275.41006

Summary: Using rational functions of the form \[ \left\{\frac{t^{12m}(t-(2-\sqrt3))^{12m}}{1+t^2}\right\}_{m\in\mathbb N} \] we produce a family of efficient polynomial approximations to arctangent on the interval \([0,2-\sqrt3]\), and hence, provide approximations to \(\pi\) via the identity \(\arctan(2-\sqrt3)=\pi/12\). We turn the approximations of \(\pi\) into a series that gives about 21 more decimal digits of accuracy with each successive term.

MSC:

41A10 Approximation by polynomials
26D05 Inequalities for trigonometric functions and polynomials
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