The enhancement of Machin’s formula by Todd’s process

Abstract

1. BackgroundFrom the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind isWriting x = cotθ, the series becomesExpressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’: