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O[x]^n

represents a term of order xn.

O[x]^n

is generated to represent omitted higher‐order terms in power series.

O[x,x0]^n

represents a term of order (x-x0)n.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
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O

O[x]^n

represents a term of order xn.

O[x]^n

is generated to represent omitted higher‐order terms in power series.

O[x,x0]^n

represents a term of order (x-x0)n.

Details

  • Normal can be used to truncate power series, and remove O terms.

Examples

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Basic Examples  (1)

A series representing an approximate function:

Wolfram Language code: x + x ^ 2 / 2 + O[x] ^ 10

Doing operations on the approximate function maintains the appropriate series order:

Wolfram Language code: 1 / (1 + Exp[%])

Scope  (2)

Like approximate numbers, approximate functions are contagious:

Wolfram Language code: Exp[x] + O[x] ^ 6
Wolfram Language code: Pi ^ 2 + 1.3

O can be used to drop higher-order terms:

Wolfram Language code: 1 + x ^ 10 + O[x] ^ 5

Generalizations & Extensions  (3)

O[x,x0] represents a term with expansion point x0:

Wolfram Language code: Exp[x] + O[x, 1] ^ 5

With expansion point ∞, the effective variable is :

Wolfram Language code: 1 / (1 + x) ^ 2 + O[x, Infinity] ^ 10

An expression involving O is converted to a SeriesData object:

Wolfram Language code: InputForm[x + x ^ 2 / 2 + O[x] ^ 10]

Applications  (1)

Find the lowest-order terms in a large polynomial:

Wolfram Language code: (1 + 2x) ^ 1000 + O[x] ^ 5
Wolfram Language code: Normal[%]

Properties & Relations  (1)

Applying a function may give a different number of terms than generating the series from scratch:

Wolfram Language code: Sin[x ^ 2 + x ^ 3 + O[x] ^ 10]
Wolfram Language code: Series[Sin[x ^ 2 + x ^ 3], {x, 0, 10}]
Wolfram Research (1988), O, Wolfram Language function, https://reference.wolfram.com/language/ref/O.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_o, author="Wolfram Research", title="{O}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/O.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_o, organization={Wolfram Research}, title={O}, year={1988}, url={https://reference.wolfram.com/language/ref/O.html}, note=[Accessed: 12-June-2026]}