Sequence and Series > Arctangent Series Expansion

## What is an Arctangent Series Expansion?

The arctangent function can be expanded as a Maclaurin series:

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

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) [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

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