In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics.^[1] Using modern notation, these series are:

${\displaystyle {\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}\theta ^{2k+1},\\\cos \theta &=1-{\frac {\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}\theta ^{2k},\\\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}|x|\leq 1.\end{alignedat}}}$

All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669,^[2] and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673,^[3] and is conventionally called Gregory's series. The specific value ${\textstyle \arctan 1={\tfrac {1}{4}}\pi }$ 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,^[4] Madhava–Gregory series,^[5] or Madhava–Leibniz series^[6] (among other combinations).^[7]

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 attributions of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series.

Madhava series in "Madhava's own words"

None of Madhava's works, containing any of the series expressions attributed to him, have survived. These series expressions are found in the writings of the followers of Madhava in the Kerala school. At many places these authors have clearly stated that these are "as told by Madhava". Thus the enunciations of the various series found in Tantrasamgraha and its commentaries can be safely assumed to be in "Madhava's own words". 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.^[8][9]

Madhava's sine series

In Madhava's own words

Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.

Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide by the squares of the successive even numbers (such that current is multiplied by previous) increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva , as collected together in the verse beginning with "vidvan" etc.

Rendering in modern notations

Let r denote the radius of the circle and s the arc-length.

• The following numerators are formed first:

  ${\displaystyle s\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad \cdots }$

• These are then divided by quantities specified in the verse.

  ${\displaystyle s\cdot <!--\; \cdot \; -->\frac{s^2}{(2^2+2)r^2},s\cdot <!--\; \cdot \; -->\frac{s^2}{(2^2+2)r^2}\cdot <!--\; \cdot \; -->\frac{s^2}{(4^2+4)r^2},s\cdot <!--\; \cdot \; -->\frac{s^2}{(2^2+2)r^2}\cdot <!--\; \cdot \; -->\frac{s^2}{(4^2+4)r^2}\cdot <!--\; \cdot \; -->\frac{s^2}{(6^2+6)r^2}}$Zdroj:https://en.wikipedia.org?pojem=Madhava_series
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