How did Madhava come up with the Arctan series?

[imurme8]

Does anyone know how Madhava discovered the power series for the arctangent? I think the standard way is to note that $1-x^2+x^4-\dotsb$ converges uniformly on $(-1,1)$ to $\frac{d}{dt}\tan^{-1}x$, and thus applying the fundamental theorem of calculus we may integrate term-by-term. But how did Madhava do it? I don't know that he had the FTOC or a concept of uniform convergence, or even that he knew how to integrate a polynomial.

 

[mathwonk]

obviously it was revealed by vishnu.

 

[Gravitational]

obviously it was revealed by vishnu.

That made me laugh.

 

[arildno]

The problem with much of Indian mathematics is that it is in the form: "See!", with no attendant explanations.
Presumably, such explanations were developed in the local schools and research centres, but we have not, unfortunately, been handed down lecture notes and such from those days.

If I were to make a guess, I would think they were well aware of the sum of an infinite geometric series, and that the result given has a close relationship to their understanding of how 1/(1+x^2) appeared within trigonometry.

 

[Mute]

The wikipedia articles about him say that his methods are explained in a book by some of his followers. The articles briefly describe his methods:

Overview of Medhava

Wiki article about madhava series