Gregory's series

In mathematics, Gregory's series for the inverse tangent function is its infinite Taylor series expansion at the origin:

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

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 Madhava of Sangamagrama (c. 1340 – c. 1425), as credited by Madhava's Kerala school follower Jyeṣṭhadeva's Yuktibhāṣā (c. 1530). In recent literature it is sometimes called the Madhava–Gregory series to recognize Madhava's priority (see also Madhava series). It was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673, who obtained the Leibniz formula for π as the special case

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