**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 a_{n}n^{-s}=: f(s) ∈ ℂ. 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.

## Common Series Expansions

## References

Indiana University Bloomington. (2020). Standards for Reporting the Optical Aberrations of Eyes.

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

**CITE THIS AS:**

**Stephanie Glen**. "Series Expansion: Definition, Common Types" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/series-expansion/

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