Power Series Expansions in India Around A. D. 1400

Abstract

Mādhava of San$\dot{\rm n}$gamagrāma was a mathematical astronomer who flourished around 1400 A.D. in South India. Not a few mathematical formulas attributed to him have been transmitted by scholars of his school to these days. The most important among them are power series expansions of trigonometrical functions sine, cosine, arctangent, and so on. Because these series are not found in his extant astronomical treatises it is not always clear what parts of them were found by Mādhava himself, though some, the expansion of the circumference of a circle for example, can be attributed to him with a certainty.

After taking a glance at Mādhava himself and his school, I will explain in this paper howMādhava derived the power series of the circumference C of a circle with diameter d together with the last corrective term:

$$ C= \frac{4d}{1}-\frac{4d}{3}+\frac{4d}{5}-\frac{4d}{7}+\cdots+(-1)^{n-1} \frac{4d}{2n-1}+(-1)^n\cdot 4d \cdot\frac{n}{(2n)^2+1}, $$

according to the commentary Kriyākramakarī (ca. AD 1550), composed by Śa$\dot{\rm n}$kara, who was a scholar situated near the end of the Mādhava school.