Madhava of Sangamagrama is credited with founding the Kerala School of Astronomy and Mathematics, which made significant contributions to mathematics, particularly in the area of series expansions.

Madhava of Sangamagrama developed a series expansion for the sine function, known as Madhava's Sine Series, which is considered a precursor to the modern Taylor Series.

Madhava of Sangamagrama developed series expansions for various trigonometric functions, including sine, cosine, and tangent.

Madhava's series expansions had far-reaching implications in the development of calculus, contributing to the understanding of trigonometric functions, limits, derivatives, and integrals.

Madhava's work on series expansions was rediscovered in the 17th century by the Scottish mathematician John Gregory, who published it in his book 'Vera Circuli et Hyperbolae Quadratura'.

Madhava of Sangamagrama also developed a series expansion for the arctangent function, known as Madhava's Arctangent Series.

Madhava of Sangamagrama developed a series expansion for the arctangent function, which is given by $$\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$. This series converges for (|x| \leq 1).

Madhava's series expansions had a profound impact on the development of modern mathematics, providing a basis for calculus, advancing the study of infinite series and convergence, and inspiring the work of subsequent mathematicians.

Madhava of Sangamagrama developed a series expansion for the cosine function, which is given by $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$. This series converges for all (x).

Madhava of Sangamagrama also developed a series expansion for the tangent function, known as Madhava's Tangent Series.

Madhava of Sangamagrama developed a series expansion for the tangent function, which is given by $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$. This series converges for (|x| \leq \frac{\pi}{4}).

Madhava of Sangamagrama's work on series expansions was driven by a combination of motivations, including the desire to accurately calculate trigonometric ratios, explore the nature of infinite series, and solve problems in astronomy and mathematics.

Madhava of Sangamagrama developed a series expansion for the sine function, which is given by $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$. This series converges for all (x).

Madhava of Sangamagrama also developed a series expansion for the arccosine function, known as Madhava's Arccosine Series.

Madhava of Sangamagrama developed a series expansion for the arccosine function, which is given by $$\cos^{-1} x = \frac{\pi}{2} - x + \frac{x^3}{2\cdot3} - \frac{x^5}{2\cdot4\cdot5} + \frac{x^7}{2\cdot4\cdot6\cdot7} + \cdots$$. This series converges for (|x| \leq 1).