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Mathematics LibreTexts

Series and Expansions

A series is the sum of the terms of a finite or infinite sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. A series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting series often can be limited to a finite number of terms, thus yielding an approximation of the function. Examples include:

  • Taylor series: A power series based on a function’s derivatives at a single point.
  • Maclaurin series: A special case of a Taylor series, centerd at zero.
  • Laurent series: An extension of the Taylor series, allowing negative exponent values.
  • Dirichlet series: Used in number theory.
  • Fourier series: Describes periodical functions as a series of sine and cosine functions.
  • Newtonian series
  • Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.
  • Zernike polynomials: Used in optics to calculate aberrations of optical systems.
  • Stirling series: Used as an approximation for factorials.

Thumbnail: As the degree of the Taylor series rises, it approaches the correct function. Image used with permission (CC AS-BY 3.0; Maksim).