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Madhava series — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Madhava_series
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T14:03:29.747Z
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Madhava series — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Madhava_series (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_b03e1059cd47 (Metraipolis archive dc_b03e1059cd47, sha256 d74363d0e8aefb3fb129ab00651ef0879320d39f02becc0bbca34080e6f42402)

BibTeX

@misc{dcb03e1059cd47,
  title = {Madhava series — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Madhava_series},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_b03e1059cd47; sha256 d74363d0e8aefb3fb129ab00651ef0879320d39f02becc0bbca34080e6f42402},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Madhava_series. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, the Maclaurin series of the sine, cosine, and arctangent functions are sometimes called Madhava series because they were discovered in the 14th or 15th century by the Indian mathematician Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are sin θ = θ − θ 3 3 ! + θ 5 5 ! − θ 7 7 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos θ = 1 − θ 2 2 ! + θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! θ 2 k , arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 x 2 k + 1 where | x | ≤ 1. {\displaystyle {\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}\theta ^{2k+1},\\[10mu]\cos \theta &=1-{\frac {\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}\theta ^{2k},\\[10mu]\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}|x|\leq 1.\end{alignedat}}} All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669, and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673, and is conventionally called Gregory's series. The specific value arctan 1 = π 4 {\textstyle \arctan 1={\tfrac {\pi }{4}}} can be used to calculate the circle constant π, and the arctangent series for 1 is conventionally called Leibniz's series. In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations). No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later Kerala school mathematicians Nilakantha Somayaji (1444 – 1544) and Jyeshthadeva (c. 1500 – c. 1575) one can find unambiguous attribution of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series. The translations of the relevant verses as given in the Yuktidipika commentary of Tantrasamgraha (also known as Tantrasamgraha-vyakhya) by Sankara Variar (circa. 1500 - 1560 CE) are reproduced below. These are then rendered in current mathematical notations.

Tema degli esploratori: storia e filosofia della scienza (deposito sistematico; categoria «History of mathematics»).
Fonte: Wikipedia in inglese, voce «Madhava series» — https://en.wikipedia.org/wiki/Madhava_series
Wikidata Q6727285 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q6727285
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Madhava_series?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1375245198, consultata il 2026-10-07T14:00:47.448Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 741e66711fe7cef6dde0e63b7ebfcdc54887f763a5dcb3d826cc1261609fa4f0