The π Series Revisited: Algebra in Analysis

Abstract

In this final chapter on the mathematics of the Nila school, I return to a topic which was put aside earlier. The problem which it deals with has its origin in the realisation of the extremely slow convergence of the basic π series, making it effectively useless for the calculation of good approximations to its numerical value. Having recognised the problem, Madhava set about overcoming it in an exercise as remarkable for its originality and effectiveness as for its sense of modernity. Two closely related ways of dealing with the slowness of convergence emerge: i) a sequence of better and better estimates, potentially without end, for the remainder in the basic π series when it is terminated after a finite but arbitrary number of terms, as rational functions of the termination point; and ii) reorganisations of the series, based on these ‘error’ estimates, to produce new series that converge more rapidly.