The infinite series for π was stated by Madhava of Sangamagrama (c. 1340-1425) and his Kerala school of astronomy and mathematics. He made use of the series expansion of arctanx to obtain an infinite series expression, now known as the Madhava-Gregory series, for π. Their rational approximation of the error for the finite sum of their series are of particular interest. They manipulated the error term to derive a faster converging series for π. They used the improved series to derive a rational expression, 104348 / 33215 for π correct up to nine decimal places, i.e. 3.141592653
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