Gregory's series
Introduction In mathematics, '''Gregory's series''' for the inverse tangent function is its infinite Taylor series expansion at the origin::This series converges in the complex disk  except for  (where 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  as the special case:
Proof The derivative of  is ; conversely, the integral of  is .If  then  The derivative is:Taking the reciprocal,:This sometimes is used as a definition of the arctangent::The Maclaurin series for  is a geometric series::One can find the Maclaurin series for  by naïvely integrating term-by-term::While this turns out correctly, integrals and infinite sums cannot always be exchanged in this manner. To prove that the integral on the left converges to the sum on the right for real   can instead be written as the ''finite'' sum,:Again integrating both sides, :In the limit as  the integral on the right above tends to zero when  because:Therefore,:
Convergence The series for  and  converge within the complex disk , where both functions are holomorphic. They diverge for  because when , there is a pole::When  the partial sums  alternate between the values  and  never converging to the value However, its term-by-term integral, the series for  (barely) converges when  because  disagrees with its series only at the point  so the difference in integrals can be made arbitrarily small by taking sufficiently many terms: :Because of its exceedingly slow convergence (it takes five billion terms to obtain 10 correct decimal digits), the Leibniz formula is not a very effective practical method for computing  Finding ways to get around this slow convergence has been a subject of great mathematical interest.
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  include Nilakantha Somayaji's Tantrasangraha (c. 1500), Jyeṣṭhadeva's ''Yuktibhāṣā'' (c. 1530), and the ''Yukti-dipika'' commentary by Sankara Variyar, where it is given in verses 2.206 – 2.209.
See also * List of mathematical series* Madhava series
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