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SeriesData
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SeriesData

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SeriesData[x,x0,{a0,a1,…},nmin,nmax,den]

represents a power series in the variable x about the point x0. The ai are the coefficients in the power series. The powers of (x-x0) that appear are nmin/den, (nmin+1)/den, …, nmax/den.

Details

  • SeriesData objects are generated by Series.
  • SeriesData objects are printed as sums of the coefficients ai, multiplied by powers of x-x0. A SeriesData object representing a power series is printed with O[x-x0]^p added, to represent omitted higher‐order terms.
  • When you apply certain mathematical operations to SeriesData objects, new SeriesData objects truncated to the appropriate order are produced.
  • The operations you can perform on SeriesData objects include arithmetic ones, mathematical functions with built‐in derivatives, and integration and differentiation.
  • Normal[expr] converts a SeriesData object into a normal expression, truncating omitted higher‐order terms.
  • If the variable in a SeriesData object is itself a SeriesData object, then the composition of the SeriesData objects is computed. Substituting one series into another series with the same expansion parameter therefore automatically leads to composition of the series. Composition is only possible if the first term of the inner series involves a positive power of the variable.
  • InverseSeries can be applied to SeriesData objects to give series for inverse functions.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

SeriesData is the head of the basic expressions generated by Series:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-rn0
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdoubqzm-xfm

Normal converts to an ordinary expression:

In[3]:=3
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https://wolfram.com/xid/0bdoubqzm-k5t
Out[3]=3

Scope  (5)Survey of the scope of standard use cases

Construct SeriesData explicitly:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-um5
Out[1]=1

Rational powers are directly accounted for in SeriesData:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-x83
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdoubqzm-dml

Logarithms or other functions increasing slower than powers are explicitly included as coefficients:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-y2p
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdoubqzm-ff7

Symbolic powers are kept outside SeriesData:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-ucr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdoubqzm-f8p

Rapidly increasing or oscillating functions are kept outside SeriesData:

In[1]:=1
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https://wolfram.com/xid/0bdoubqzm-u2v
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdoubqzm-nqy
Wolfram Research (1988), SeriesData, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesData.html.
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Wolfram Research (1988), SeriesData, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesData.html.

Text

Wolfram Research (1988), SeriesData, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesData.html.

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Wolfram Research (1988), SeriesData, Wolfram Language function, https://reference.wolfram.com/language/ref/SeriesData.html.

CMS

Wolfram Language. 1988. "SeriesData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SeriesData.html.

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Wolfram Language. 1988. "SeriesData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SeriesData.html.

APA

Wolfram Language. (1988). SeriesData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SeriesData.html

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Wolfram Language. (1988). SeriesData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SeriesData.html

BibTeX

@misc{reference.wolfram_2025_seriesdata, author="Wolfram Research", title="{SeriesData}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/SeriesData.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_seriesdata, author="Wolfram Research", title="{SeriesData}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/SeriesData.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_seriesdata, organization={Wolfram Research}, title={SeriesData}, year={1988}, url={https://reference.wolfram.com/language/ref/SeriesData.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_seriesdata, organization={Wolfram Research}, title={SeriesData}, year={1988}, url={https://reference.wolfram.com/language/ref/SeriesData.html}, note=[Accessed: 29-March-2025 ]}