4. In the 14th century, Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, used the usu

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4 In The 14th Century Madhava Of Sangamagrama Founder Of The Kerala School Of Astronomy And Mathematics Used The Usu 1 (72.1 KiB) Viewed 38 times

4. In the 14th century, Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, used the usual infinite series for arctan x to approximate n. In this exercise we'll reproduce their method and determine how accurate the approximation is. The key idea is to notice that arctan(1/3) = 7/6 (i) Write down the first 8 nonzero terms of the Taylor series of arctan x and evaluate it at 1/ 13. (ii) The error of this approximation of 7/6 is bounded by the absolute value of the 9th term of the Taylor series evaluated at 1/13. Compute this term and estimate the error in Madhava's approximation.