Calculus How To

Series Expansion: Definition, Common Types

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What is a Series Expansion?

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

For example, the natural exponential function ex can be expanded into an infinite series:
e series expansion

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:

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: arctan series expansion
  • Dirichlet series: Any series of the form dirichlet seriesan 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

common series expansion part 1

common series expansions


Indiana University Bloomington. (2020). Standards for Reporting the Optical Aberrations of Eyes.
McCarthy, J. (2018). Dirichlet Series, Retrieved December 2, 2020 from:

Stephanie Glen. "Series Expansion: Definition, Common Types" From Calculus for the rest of us!

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