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GeezerVison is showing reference.wolfram.com — Infinity—Wolfram Documentation Infinity or is a symbol that represents a positive infinite quantity. Details
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InfinityInfinity or is a symbol that represents a positive infinite quantity. Details
Examplesopen all close allBasic Examples (2)Use as iterator limit: Sum[1 / n ^ 2, {n, Infinity}]Do arithmetic with infinite quantities: 1 / InfinityScope (7)Use Limit[1 / x, x -> ∞]Series expansion at Series[1 / (x ^ 2 + x + 1), {x, Infinity, 5}]Infinite limits of integration and summation: Integrate[Exp[-x ^ 2], {x, -Infinity, Infinity}]NIntegrate[Exp[-x ^ 4 / (x ^ 2 + 1)], {x, -Infinity, Infinity}]Sum[Exp[-n ^ 2], {n, -Infinity, Infinity}]NSum[Exp[-n ^ 4 / (n ^ 2 + 1)], {n, -Infinity, Infinity}]Extended arithmetic with infinite quantities: Infinity Infinity2 Infinity - Infinity
In this case the result depends on the growth rates of the two terms: Limit[2t - (t ^ 2 + 1), t -> Infinity]Limit[2t - (t + 1), t -> Infinity]Use infinite quantities in predicates: 10 ^ 100 < InfinityFind values of mathematical functions at infinities: Exp[-Infinity]Exp[Infinity]For some functions there is no definite limit, but a limiting interval: Sin[Infinity]Tan[Infinity]In this case no limiting interval exists: Exp[I Infinity]
Mathematical functions can also return infinite quantities at finite points: Log[0]Gamma[0]Cases finds matches only at the first level by default: e = x Gamma[x] + (x + 1)Gamma[x ^ 2 + 1]Cases[e, _Gamma]Set the level to Infinity to include all levels: Cases[e, _Gamma, Infinity]Applications (2)Stirling's asymptotic approximation: Series[n!, {n, Infinity, 2}]Setting the precision to Infinity will produce a rational approximation: SetPrecision[3.14, Infinity]N[%]Some functions require infinite precision or exact input: CylindricalDecomposition[y ^ 2 + 2.5 x ^ 2 < 1.5, {x, y}]
Apply SetPrecision to the whole expression to rationalize all inexact numbers: CylindricalDecomposition[SetPrecision[y ^ 2 + 2.5 x ^ 2 < 1.5, Infinity], {x, y}]Neat Examples (1)Use an infinite WorkingPrecision in Plot: Reap[Plot[2x + 1, {x, 0, 1}, WorkingPrecision -> Infinity, EvaluationMonitor :> Sow[x], PlotPoints -> 3, MaxRecursion -> 0 ]]Plot[Numerator[x], {x, 0, 10}, WorkingPrecision -> Infinity, PlotPoints -> 1000, MaxRecursion -> 0 ]Related GuidesRelated LinksHistoryIntroduced in 1988 (1.0) | Updated in 1996 (3.0)
Wolfram Research (1988), Infinity, Wolfram Language function, https://reference.wolfram.com/language/ref/Infinity.html (updated 1996).
TextWolfram Research (1988), Infinity, Wolfram Language function, https://reference.wolfram.com/language/ref/Infinity.html (updated 1996). CMSWolfram Language. 1988. "Infinity." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Infinity.html. APAWolfram Language. (1988). Infinity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Infinity.html BibTeX@misc{reference.wolfram_2026_infinity, author="Wolfram Research", title="{Infinity}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Infinity.html}", note=[Accessed: 12-November-2026]} BibLaTeX@online{reference.wolfram_2026_infinity, organization={Wolfram Research}, title={Infinity}, year={1996}, url={https://reference.wolfram.com/language/ref/Infinity.html}, note=[Accessed: 12-November-2026]} |
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