Series [f,{x,x0,n}]
generates a power series expansion for f about the point x=x0 to order (x-x0)n, where n is an explicit integer.
Series [f,xx0]
generates the leading term of a power series expansion for f about the point x=x0.
Series [f,{x,x0,nx},{y,y0,ny},…]
successively finds series expansions with respect to x, then y, etc.
Series
Series [f,{x,x0,n}]
generates a power series expansion for f about the point x=x0 to order (x-x0)n, where n is an explicit integer.
Series [f,xx0]
generates the leading term of a power series expansion for f about the point x=x0.
Series [f,{x,x0,nx},{y,y0,ny},…]
successively finds series expansions with respect to x, then y, etc.
Details and Options
- Series can construct standard Taylor series, as well as certain expansions involving negative powers, fractional powers, and logarithms.
- Series detects certain essential singularities. On [Series::esss] makes Series generate a message in this case.
- Series can expand about the point x=∞.
- Series [f,{x,0,n}] constructs Taylor series for any function f according to the formula .
- Series effectively evaluates partial derivatives using D . It assumes that different variables are independent.
- The result of Series is usually a SeriesData object, which you can manipulate with other functions.
- Normal [series] truncates a power series and converts it to a normal expression.
- SeriesCoefficient [series,n] finds the coefficient of the n^(th)-order term.
- The following options can be given:
-
Examples
open all close allBasic Examples (4)
Power series for the exponential function around :
Convert to a normal expression:
Power series of an arbitrary function around :
In any operation on series, only appropriate terms are kept:
Find the leading term of a power series:
Scope (10)
Univariate Series (10)
Series can handle fractional powers and logarithms:
Symbolic parameters can often be used:
Laurent series with negative powers can be generated:
Truncate the series to the specified negative power:
Find power series for special functions:
Find the series for a function at a branch point:
With x assumed to be to the left of the branch point, a simpler result is given:
Piecewise functions:
Power series at infinity:
Series can give asymptotic series:
Series expansions of implicit solutions to equations:
Series expansions of unevaluated integrals:
Generalizations & Extensions (4)
Power series in two variables:
Series is threaded element-wise over lists:
Series generates SeriesData expressions:
Series can work with approximate numbers:
Options (4)
Analytic (1)
Series by default assumes symbolic functions to be analytic:
Assumptions (3)
Use Assumptions to specify regions in the complex plane where expansions should apply:
Without assumptions, piecewise functions appear:
Get expansions in Stokes regions:
Applications (8)
Plot successive series approximations to :
Find a series expansion for a standard combinatorial problem:
Find Fibonacci numbers from a generating function:
Find Legendre polynomials by expanding a generating function:
Set up a generating function to enumerate ways to make change using U.S. coins:
The number of ways to make change for 1ドル:
Find the lowest-order terms in a large polynomial:
Find higher-order terms in Newton's approximation for a root of f[x] near :
Plot the complex zeros for a series approximation to Exp [x]:
Properties & Relations (10)
Series always only keeps terms up to the specified order:
Operations on series keep only the appropriate terms:
Normal converts to an ordinary polynomial:
Any mathematical function can be applied to a series:
Adding a series of lower order causes the higher-order terms to be dropped:
Differentiate a series:
Solve equations for series coefficients:
Find the list of coefficients in a series:
Use O [x] to force the construction of a series:
ComposeSeries treats a series as a function to apply to another series:
InverseSeries does series reversion to find the series for the inverse function of a series:
Use FunctionAnalytic to test whether a function is analytic:
An analytic function can be expressed as a Taylor series at each point of its domain:
The resulting polynomial approximates near 0:
Possible Issues (7)
When there is an essential singularity, Series will attempt to factor it out:
Numeric values cannot be substituted directly for the expansion variable in a series:
Use Normal to get a normal expression in which the substitution can be done:
Series must be converted to normal expressions before being plotted:
Power series with different expansion points cannot be combined:
Not all series are represented by expressions with head SeriesData :
Some functions cannot be decomposed into series of power-like functions:
Series does not change expressions independent of the expansion variable:
Related Guides
Related Links
History
Introduced in 1988 (1.0) | Updated in 2020 (12.1)
Text
Wolfram Research (1988), Series, Wolfram Language function, https://reference.wolfram.com/language/ref/Series.html (updated 2020).
CMS
Wolfram Language. 1988. "Series." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Series.html.
APA
Wolfram Language. (1988). Series. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Series.html
BibTeX
@misc{reference.wolfram_2025_series, author="Wolfram Research", title="{Series}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Series.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_series, organization={Wolfram Research}, title={Series}, year={2020}, url={https://reference.wolfram.com/language/ref/Series.html}, note=[Accessed: 16-November-2025]}