Madhava of Sangamagrama and His Series Expansions

Madhava of Sangamagrama is considered the founder of the Kerala school of astronomy and mathematics, which flourished in the 14th and 15th centuries.

The Madhava series is a series expansion that approximates the sine function. It is named after Madhava of Sangamagrama, who discovered it in the 14th century.

The general formula for the Madhava series is $$sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$. This series converges for all values of x.

The Madhava series is a series expansion that approximates the cosine function. It is named after Madhava of Sangamagrama, who discovered it in the 14th century.

The general formula for the Madhava series for the cosine function is $$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$. This series converges for all values of x.

The Madhava series is a series expansion that approximates the arctangent function. It is named after Madhava of Sangamagrama, who discovered it in the 14th century.

The general formula for the Madhava series for the arctangent function is $$arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$. This series converges for all values of x between -1 and 1.

The Madhava series is a series expansion that approximates the pi. It is named after Madhava of Sangamagrama, who discovered it in the 14th century.

The general formula for the Madhava series for pi is $$\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$. This series converges very slowly, but it was the first series expansion for pi to be discovered.

Madhava of Sangamagrama's main contribution to mathematics was the discovery of the Madhava series, which are series expansions for the sine, cosine, arctangent, and pi functions.

Madhava of Sangamagrama lived in the 14th century.

Madhava of Sangamagrama wrote a book called The Yuktibhasa, which contains his mathematical discoveries.

The Madhava theorem states that the sum of the squares of the first n natural numbers is equal to \frac{n(n+1)(2n+1)}{6}.

The Madhava theorem states that the area of a circle is equal to \frac{\pi r^2}{2}.