Arctangent series

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:^[1]

${\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{2k+1}}.}$

This series converges in the complex disk ${\displaystyle |x|\leq 1,}$ except for ${\displaystyle x=\pm i}$ (where ${\displaystyle \arctan \pm i=\infty }$).

It was first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma (c. 1340 – c. 1425), the founder of the Kerala school, and is described in extant works by Nīlakaṇṭha Somayāji (c. 1500) and Jyeṣṭhadeva (c. 1530). Mādhava's work was unknown in Europe, and the arctangent series was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673.^[2] In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series).^[3]

The special case of the arctangent of ⁠${\displaystyle 1}$⁠ is traditionally called the Leibniz formula for π, or recently sometimes the Mādhava–Leibniz formula:

${\displaystyle {\frac {\pi }{4}}=\arctan 1=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots .}$

The extremely slow convergence of the arctangent series for ${\displaystyle |x|\approx 1}$ makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed ⁠${\displaystyle {\tfrac {1}{4}}\pi }$⁠ as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for ⁠${\displaystyle \pi }$⁠. Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.