On a new type of rational and highly convergent series, by which the ratio of the circumference to the diameter is able to be expressed
arXiv:math/0508170
Abstract
In this short paper Euler gives a highly convergent series for arctan and thus pi, which converges much faster than the Leibniz series for arctan.
5 pages, seems to be first English translation of Euler's Latin original ``De novo genere serierum rationalium et valde convergentium, quibus ratio peripheriae ad diametrum exprimi potest'', Nova Acta Academiae Scientarum Imperialis Petropolitinae 11 (1798), 150-154, E706