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In mathematics, a Madhava series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics. These expressions are the infinite power series expansions of the trigonometric sine, cosine and arctangent functions, and the special case of the power series expansion of the arctangent function yielding a formula for computing π. The power series expansions of sine and cosine functions are respectively called ''Madhava's sine series'' and ''Madhava's cosine series''. The power series expansion of the arctangent function is sometimes called ''Madhava–Gregory series''〔Reference to Gregory–Madhava series: 〕〔Reference to Gregory–Madhava series: 〕 or ''Gregory–Madhava series''. These power series are also collectively called ''Taylor–Madhava series''. The formula for π is referred to as ''Madhava–Newton series'' or ''Madhava–Leibnitz series'' or Leibniz formula for pi or Leibnitz–Gregory–Madhava series. These further names for the various series are reflective of the names of the Western discoverers or popularizers of the respective series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in infinite series and the use of a base ten decimal system also suggest that it was possible for calculus to have developed in India almost 300 years before its recognized birth in Europe. No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later members of the Kerala school of astronomy and mathematics like Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. It is also in the works of these later astronomers and mathematicians one can trace the Indian proofs of these series expansions. These proofs provide enough indications about the approach Madhava had adopted to arrive at his series expansions. Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions. But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for π (this was two centuries before Leibniz was to come to the same conclusion in Europe). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places. 〔(【引用サイトリンク】url=http://www.storyofmathematics.com/indian_madhava.html )〕 ==Madhava's series in modern notations== In the writings of the mathematicians and astronomers of the Kerala school, Madhava's series are described couched in the terminology and concepts fashionable at that time. When we translate these ideas into the notations and concepts of modern day mathematics, we obtain the current equivalents of Madhava's series. These present-day counterparts of the infinite series expressions discovered by Madhava are the following: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Madhava series」の詳細全文を読む スポンサード リンク
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