# Gregory's series

Gregory's series, also known as the Madhava–Gregory series or Leibniz's series, is an infinite Taylor series expansion of the inverse tangent function. It was discovered by the Indian mathematician Madhava of Sangamagrama (1350–1410) (Gupta 1973) and rediscovered in 1668 by James Gregory. It was re-rediscovered a few years later by Gottfried Leibniz, who reobtained the Leibniz formula for π (which had also been obtained earlier by Madhava) as the special case x = 1 of the Gregory series.[1]

## The series

The series is,

${\displaystyle \int _{0}^{x}\,{\frac {du}{1+u^{2}}}=\arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots .}$

Compare with the series for sine, which is similar but has factorials in the denominator.

## History

The earliest person to whom the series can be attributed with confidence is Madhava of Sangamagrama (c. 1340 – c. 1425). The original reference (as with much of Madhava's work) is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for arctanθ include Nilakantha Somayaji's Tantrasangraha (c. 1500),[2]

[3]

Jyeṣṭhadeva's Yuktibhāṣā (c. 1530),[4] and the Yukti-dipika commentary by Sankara Variyar, where it is given in verses 2.206 – 2.209.[5]

Gregory is cited for the series based on two publications in 1668, Geometriae pars universalis (The Universal Part of Geometry), Exercitationes geometrica (Geometrical Exercises).