Gregory's series, is an infinite

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

expansion of the

inverse tangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Sp ...

function. It was discovered in 1668 by James Gregory. It was re-rediscovered a few years later by

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

, who re obtained the

Leibniz formula for π In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general serie ...

as the special case ''x'' = 1 of the Gregory series.

The series

The series is, :

\int_0^x \, \frac =  \arctan x = x - \frac + \frac - \frac + \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 Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is ...

(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 The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...

founded by him. Specific citations to the series for arctanθ include

Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensi ...

Tantrasangraha Tantrasamgraha, or Tantrasangraha, (literally, ''A Compilation of the System'') is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The ...

(c. 1500),

Jyeṣṭhadeva Jyeṣṭhadeva (Malayalam: ജ്യേഷ്ഠദേവൻ) () was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama (). He is best known as the author of '' Yuktibhāṣā'', a ...

's ''

Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as Gaṇita-yukti-bhāṣā and (''Compendium of Astronomical Rationale''), is a major treatise on Indian mathematics, mathematics and Hindu astronomy, astronomy, ...

'' (c. 1530), and the ''Yukti-dipika'' commentary by

Sankara Variyar Shankara Variyar (; .) was an astronomer-mathematician of the Kerala school of astronomy and mathematics. His family were employed as temple-assistants in the temple at near modern Ottapalam. Mathematical lineage He was taught mainly by Nilakan ...

, where it is given in verses 2.206 – 2.209. Gregory is cited for the series based on two publications in 1668, ''Geometriae pars universalis'' (The Universal Part of Geometry), ''Exercitationes geometrica'' (Geometrical Exercises).

References

* Carl B. Boyer, A history of mathematics, 2nd edition, by John Wiley & Sons, Inc., page 386, 1991 * {{cite journal, first=RC, last=Gupta, title=The Madhava–Gregory series, journal=Mathematical Education, volume=7, year=1973, pages=67–70 Mathematical series

The series

History

See also

References