---
tag: QF83S57V8Y
kind: concept
name: Madhava series
aliases: [madhava-cosine-series, madhava-sine-series, madhavas-arctangent-series, madhavas-cosine-series, madhavas-series, madhavas-sine-series]
sources:
  - wikipedia: en:26149716@r1375245198
    locator: Madhava series
    note: adopted
    mention: wikipedia://en/26149716
    sections: [Summary]
same_as:
  wikidata: Q6727285
  wikipedia: Madhava series
inventory: [SGYVTGTR75]
---

## Summary

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

$$\begin{alignedat}{3} \sin \theta &= \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots &&= \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}\theta^{2k+1}, \\[10mu] \cos \theta &= 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots &&= \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}\theta^{2k}, \\[10mu] \arctan x &= x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots &&= \sum_{k=0}^\infty \frac{(-1)^k}{2k+1}x^{2k+1} \quad \text{where } |x| \leq 1. \end{alignedat}$$