On the acceleration of the convergence of the "M\=adhava-Leibniz series"

TL;DR

This paper explores historical Indian mathematical series related to circle calculations, highlighting innovative convergence acceleration techniques from the 14th to 16th centuries, and provides modern justification for these ancient results.

Contribution

It uncovers and mathematically justifies early Indian series expansions that converge faster than the classical Leibniz series for arctan(1).

Findings

Ancient Indian series expansions for circle calculations.

Faster convergence methods derived from continued fractions.

Modern justification of historical mathematical results.

Abstract

This paper expounds very innovative results achieved between the mid-14th century and the beginning of the 16th century by Indian astronomers belonging to the so-called "M\=adhava school". These results were in keeping with researches in trigonometry: they concern the calculation of the eight of the circumference of a circle. They not only expose an analog of the series expansion of arctan(1) usually known as the "Leibniz series", but also other analogs of series expansions, the convergence of which is much faster. These series expansions are derived from evaluations of the rests of the partial sums of the primordial series, by means of some convergents of generalized continued fractions. A justification of these results in modern terms is provided, which aims at restoring their full mathematical interest.