Sequence and Series > Arctangent Series Expansion
What is an Arctangent Series Expansion?
The arctangent series expansion is derived by taking the basic integral :
The integrand is then replaced with the series:
Finally, each term is individually integrated to give the series (for -1 < x < 1). Note that both sides equal zero when x = 0, so there’s no “+ C”.
Although the series is usually attributed to Gottfried Leibniz (1646-1716) or James Gregory (1833 to 1675) , it was known two centuries earlier to Indian mathematician Nilakantha Somayaji (ca. 1444–1544) .
Why is the Arctan Series Expansion Important?
Perhaps the most widespread us of the arctangent series is as an approximation for π. As well as the ratio of a circle’s circumference to its diameter, π is also defined as twice the least positive x for which cos(x) = 0.
Depending on the author, there are between 2 and 11 terms for the series expansion. More terms doesn’t necessarily mean more accuracy: Machin’s two term series approximates π as 3. 157866845 and Dodgson’s 11 term series gives π as 3.077143544 .
Another reason for having an interest in the arctan series is purely for historical interest. The history of this particular series is important because it was developed pre-calculus; It demonstrates early ideas on series and how they connect with quadrature or processes for finding the area under a curve (a.k.a. integration) .
 2.3 Computing Pi (continued). Retrieved April 6, 2021 from:
 Hwang Chien-Lih. (2004). Some observations on the method of arctangents for the calculation of π. The Mathematical Gazette. The Mathematical Association.
 Abeles, F. Charles L. Dodgson’s Geometric Approach to Arctangent Relations for Pi. Historia Mathematica 20, pp. 151-159. Retrieved April 6, 2021 from: http://users.uoa.gr/~apgiannop/Sources/Dodgson-pi.pdf
 Roy, R. (1990). The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha. Retrieved April 6, 2021 from: http://users.uoa.gr/~apgiannop/Sources/Roy-pi.pdf
Stephanie Glen. "Arctangent Series Expansion" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/arctangent-series-expansion/
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