A new series for $\pi$ via polynomial approximations to arctangent

Abstract

Using rational functions of the form

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

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.