**Contents:**

## What is a Series Expansion?

A **series expansion** is where a function is represented by a sum of powers of either:

- One of its variables,
- Another function (usually an elementary function).

For example, the natural exponential function e^{x} can be expanded into an infinite series:

This particular expansion is called a Taylor series.

Series expansions have a myriad of uses in a vast array of scientific areas. For example, in calculus, if you know the value of a function at a certain point (and its derivatives), you can calculate values for the whole function. Or, if you have a particularly ugly derivative or integral, you can use a series expansion to simplify the math and find an approximate solution.

## General Types of Series Expansion

The most common series expansions you’ll come across are:

- Binomial series: Two binomial quantities are raised to a power and expanded. For example, (a + b)
^{2}= (a + b) * (a + b). - Power series: Like a polynomial of infinite degree, it can be written in a few different forms. A basic example if 1 + x + x
^{2}+ … + x^{n}. - Taylor & Maclaurin Series: approximates functions with a series of polynomial functions.
- Laurent series: a way to represent a complex function as a complex power series with negative powers.

These aren’t the only tools for series expansion though. Many others exist, but **they tend to be used in very specific circumstances.** For example, Zernike polynomials are used in optics to calculate the shape of aberrated wavefronts in optical systems (Indiana, 2020) and Stirling series are used for approximating factorials. Others include:

**Arctangent series expansion**:**Dirichlet series**: Any series of the form

. The Reimann zeta function is a famous example (McCarthy, 2018).**Legendre functions of the first kind**(also called Legendre polynomials), are solutions to the Legendre differential equation.**Puiseux series**: a generalization of power series that allows for negative and fractional exponents of the indeterminate T.

## 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].

## Arctan Series Expansion: 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

## Common Series Expansions

## References

Indiana University Bloomington. (2020). Standards for Reporting the Optical Aberrations of Eyes. Retrieved July 10, 2021 from: https://www.opt.indiana.edu/vsg/library/vsia/vsia-2000_taskforce/tops4_2.html#:~:text=Standards%20for%20Reporting%20the%20Optical%20Aberrations%20of%20Eyes&text=The%20Zernike%20polynomials%20are%20a,pupil%20of%20an%20optical%20system.

McCarthy, J. (2018). Dirichlet Series, Retrieved December 2, 2020 from: https://www.math.wustl.edu/~mccarthy/amaster-ds.pdf