In
mathematics, a series expansion is a technique that expresses a
function as an infinite sum, or
series, of simpler functions. It is a method for calculating a
function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).[1]
The resulting so-called series often can be limited to a finite number of terms, thus yielding an
approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the
partial sum of the omitted terms) can be described by an equation involving
Big O notation (see also
asymptotic expansion). The series expansion on an
open interval will also be an approximation for non-
analytic functions.[2][verification needed]
Types of series expansions
There are several kinds of series expansions, listed below.
Taylor series
A Taylor series is a
power series based on a function's
derivatives at a single point.[3] More specifically, if a function is infinitely differentiable around a point , then the Taylor series of f around this point is given by
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form and converges in an
annulus.[6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Dirichlet series
Convergence and divergence of partial sums of the Dirichlet series defining the
Riemann zeta function. Here, the yellow line represents the first fifty successive partial sums the magenta dotted line represents and the green dot represents as s is varied from -0.5 to 1.5.
A Fourier series is an expansion of periodic functions as a sum of many
sine and
cosine functions.[8] More specifically, the Fourier series of a function of period is given by the expression
where the coefficients are given by the formulae[8][9]
The relative error in a truncated Stirling series vs. n, for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with
^
abEdwards, C. Henry; Penney, David E. (2008). Elementary Differential Equations with Boundary Value Problems. Pearson/Prentice Hall. p. 196.
ISBN978-0-13-600613-8.
^Weisstein, Eric W.
"Maclaurin Series". mathworld.wolfram.com. Retrieved 2022-03-22.
^Edwards, C. Henry; Penney, David E. (2008). Elementary Differential Equations with Boundary Value Problems. Pearson/Prentice Hall. pp. 558, 564.
ISBN978-0-13-600613-8.