How did Madhava’s school calculate pi to 11 decimal places?

The school of Madhava in Kerala calculated $\pi$ to 11 decimal places by utilizing infinite series expansions, specifically the power series for the arctangent function.

While the specific individual who first discovered these results is sometimes debated between Madhava and his successor Nilakantha Somayaji, the school successfully derived the series representation:

$$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$$

This was identified as a special case of the general power series for the inverse tangent function:

$${\tan }^{-1}(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots$$

Unlike European mathematicians who later utilized the fundamental theorem of calculus to derive such series, the mathematicians of the Kerala school achieved these results by finding the limits of sums of infinitesimal quantities.12