Mathematics in Ancient and Medieval India c8.s3.29

c8.s3.29

In the original series (π)/4 = 1 - ⅓ + ⅕ - 1/7 - + (1) let S ((n + 1)/2) and S ((n - 1)/2) denote sums of the first 2m + 1 and 2m terms respectively, where n = 4m + 1. These provide rational approximations to (π)/4 where n is large. Let us try to improve them by adding corrections—f(n + 1) and f(n - 1) respectively, where f(x) is a function to be suitably chosen. The improved approximations being,