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NSeries[f,{x,x0,n}]

gives a numerical approximation to the series expansion of f about the point x=x0 including the terms (x-x0)-n through (x-x0)n.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Radius  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
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NumericalCalculus`
NumericalCalculus`

NSeries

NSeries[f,{x,x0,n}]

gives a numerical approximation to the series expansion of f about the point x=x0 including the terms (x-x0)-n through (x-x0)n.

Details and Options

  • To use NSeries, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].
  • The function f must be numeric when its argument x is numeric.
  • NSeries will construct standard univariate Taylor or Laurent series.
  • NSeries samples f at points on a circle in the complex plane centered at x0 and uses InverseFourier. The option Radius specifies the radius of the circle.
  • The region of convergence will be the annulus (containing the sampled points) where f is analytic.
  • NSeries will not return a correct result if the disk centered at x0 contains a branch cut of f.
  • The result of NSeries is a SeriesData object.
  • If the result of NSeries is a Laurent series, then the SeriesData object is not a correct representation of the series, as higher-order poles are neglected.
  • No effort is made to justify the precision in each of the coefficients of the series.
  • NSeries is unable to recognize small numbers that should in fact be zero. Chop is often needed to eliminate these spurious residuals.
  • The number of sample points chosen is 2^(TemplateBox[{{{log, _, 2}, , {(, n, )}}}, Ceiling]+2).
  • The following options can be given:
  • Radius1radius of circle on which f is sampled
    WorkingPrecisionMachinePrecisionprecision used in internal computations

Examples

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Basic Examples  (1)

Wolfram Language code: Needs["NumericalCalculus`"]

This is a power series for the exponential function around :

Wolfram Language code: NSeries[Exp[x], {x, 0, 5}]

Chop is needed to eliminate spurious residuals:

Wolfram Language code: %//Chop

Using extended precision may also eliminate spurious imaginaries:

Wolfram Language code: NSeries[Exp[x], {x, 0, 5}, WorkingPrecision -> 17]

Scope  (2)

Wolfram Language code: Needs["NumericalCalculus`"]

Find expansions in the complex plane:

Wolfram Language code: NSeries[Exp[x], {x, I, 5}]//Chop
Wolfram Language code: Needs["NumericalCalculus`"]

Find Laurent expansions about essential singularities:

Wolfram Language code: NSeries[Sin[x + 1 / x], {x, 0, 10}]//Chop

Series will not find Laurent expansions about essential singularities:

Wolfram Language code: Series[Sin[x + 1 / x], {x, 0, 10}]

Options  (2)

Radius  (2)

Wolfram Language code: Needs["NumericalCalculus`"]

Use Radius to pick the annulus within which the Laurent series will converge:

Wolfram Language code: NSeries[(1/(1 + x)(3 + x)), {x, 0, 10}, Radius -> 2]//Chop

Laurent series for :

Wolfram Language code: NSeries[(1/(1 + x)(3 + x)), {x, 0, 10}, Radius -> 5]//Chop
Wolfram Language code: Needs["NumericalCalculus`"]

Changing Radius can improve accuracy:

Wolfram Language code: NSeries[(1/1 - x), {x, 0, 10}, Radius -> (7/8)]//Chop
Wolfram Language code: NSeries[(1/1 - x), {x, 0, 10}, Radius -> (1/5)]//Chop

Applications  (1)

Wolfram Language code: Needs["NumericalCalculus`"]

A function defined only for numerical input:

Wolfram Language code: f[a_ ? NumericQ] := NIntegrate[Sin[a Sin[x] + Cos[x]], {x, 0, a}]

Find a series expansion of f:

Wolfram Language code: g[a_] = Normal@NSeries[f[a], {a, 0, 10}]//Chop

Check:

Wolfram Language code: Plot[{f[a], g[a]}, {a, 0, 2}]

Properties & Relations  (1)

Wolfram Language code: Needs["NumericalCalculus`"]

NResidue can also be used to construct a series of a numerical function:

Wolfram Language code: f[x_ ? NumericQ] := Exp[x]
Wolfram Language code: NSeries[f[x], {x, 0, 5}]//Chop

Using NResidue:

Wolfram Language code: NResidue[f[x]x^Range[-1, -6, -1], {x, 0}, Radius -> 1].x^Range[0, 5]//Chop

Possible Issues  (2)

Wolfram Language code: Needs["NumericalCalculus`"]

NSeries can have aliasing problems due to InverseFourier:

Wolfram Language code: NSeries[ArcSin[x], {x, 0, 20}]//Chop

The correct expansion is analytic at the origin:

Wolfram Language code: Series[ArcSin[1.x], {x, 0, 20}]
Wolfram Language code: Needs["NumericalCalculus`"]

SeriesData cannot correctly represent a Laurent series. Here is the square of the series of Exp[+x]:

Wolfram Language code: NSeries[Exp[(1/x) + x], {x, 0, 5}]^2//Chop

Here is the SeriesData representation of the Laurent series of :

Wolfram Language code: NSeries[Exp[(1/x) + x]^2, {x, 0, 5}]//Chop

Neat Examples  (1)

Wolfram Language code: Needs["NumericalCalculus`"]

Find the series expansion of the generating function for unrestricted partitions:

Wolfram Language code: q[z_ ? NumericQ] := NProduct[(1/1 - z^n), {n, ∞}]
Wolfram Language code: NSeries[q[z], {z, 0, 20}, Radius -> (1/2), WorkingPrecision -> 17]//Chop

Check:

Wolfram Language code: PartitionsP[Range[0, 20]]
Wolfram Research (2007), NSeries, Wolfram Language function, https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html.

Text

Wolfram Research (2007), NSeries, Wolfram Language function, https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html.

CMS

Wolfram Language. 2007. "NSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html.

APA

Wolfram Language. (2007). NSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html

BibTeX

@misc{reference.wolfram_2026_nseries, author="Wolfram Research", title="{NSeries}", year="2007", howpublished="\url{https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_nseries, organization={Wolfram Research}, title={NSeries}, year={2007}, url={https://reference.wolfram.com/language/NumericalCalculus/ref/NSeries.html}, note=[Accessed: 13-June-2026]}