Nilakantha Somayaji published his work on calculus and infinite series in his book Tantrasamgraha.

Nilakantha Somayaji used a series that is now known as the Nilakantha series to calculate the value of pi.

The formula for the Nilakantha series is $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$. This series converges very quickly, and it can be used to calculate the value of pi to a high degree of accuracy.

Nilakantha Somayaji calculated the value of pi to 33 decimal places using the Nilakantha series. His value of pi is accurate to 33 decimal places.

Nilakantha Somayaji used a method that is now known as Nilakantha's method to find the derivative of a function.

The formula for Nilakantha's method for finding the derivative of a function is $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h}$$. This formula is very similar to the formula for the derivative of a function that is used in modern calculus.

Nilakantha Somayaji used a series that is now known as the Nilakantha series to find the sum of an infinite series.

The formula for the Nilakantha series for finding the sum of an infinite series is $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$. This series converges very quickly, and it can be used to find the sum of many different infinite series.

The sum of the infinite series $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$ is $$\frac{1}{2}$$. This can be shown using the Nilakantha series.

Nilakantha Somayaji used a method that is now known as the Nilakantha method to find the area of a circle.

The formula for the Nilakantha method for finding the area of a circle is $$A = \frac{1}{2}\pi r^2$$. This formula is very accurate, and it can be used to find the area of a circle to a high degree of accuracy.

The area of a circle with radius 10 units is $$A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi (10)^2 = 100\pi$$. Therefore, the area of the circle is 100\pi square units.

Nilakantha Somayaji used a method that is now known as the Nilakantha method to find the volume of a sphere.

The formula for the Nilakantha method for finding the volume of a sphere is $$V = \frac{4}{3}\pi r^3$$. This formula is very accurate, and it can be used to find the volume of a sphere to a high degree of accuracy.

The volume of a sphere with radius 10 units is $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (10)^3 = 4000\pi$$. Therefore, the volume of the sphere is 4000\pi cubic units.