Madhava's Pi series

By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary )


This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.


c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +.......


As c = Pi d , this equation can be rewritten as


Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +......


This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z).


Pi/4 = 1 - 1/3 +1/5 -1/7+.....


This is Madhava's formula for Pi, and this was discovered in the West by Gregory and Liebniz.