Calculus How To

Arctangent Series Expansion

Share on

Sequence and Series > Arctangent Series Expansion

What is an Arctangent Series Expansion?

The arctangent function can be expanded as a Maclaurin series:
arctan series expansion 1



The arctangent series expansion is derived by taking the basic integral [1]:
arctan integral

The integrand is then replaced with the series:
arctangent series expansion 3

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) [2], it was known two centuries earlier to Indian mathematician Nilakantha Somayaji (ca. 1444–1544) [2].

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 [3].

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) [4].

References

[1] 2.3 Computing Pi (continued). Retrieved April 6, 2021 from:
https://www.macalester.edu/aratra/chapt2/chapt2_3a.html
[2] Hwang Chien-Lih. (2004). Some observations on the method of arctangents for the calculation of π. The Mathematical Gazette. The Mathematical Association.
[3] 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
[4] 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

CITE THIS AS:
Stephanie Glen. "Arctangent Series Expansion" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/arctangent-series-expansion/
------------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!


Leave a Reply

Your email address will not be published. Required fields are marked *