Madhava series

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

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