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

2005/08/09 by Leonhard Euler, Euler, Leonhard
Mathematics · #Advanced Mathematical Identities #History and Theory of Mathematics #Mathematics and Applications #math.CA #math.HO #msc:01A50 #msc:40A05

paper · pdf · doi:10.48550/arxiv.math/0508170

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

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.

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