({{, (RefLink[Infinity, paclet:ref/Infinity]) or ∞
is a symbol that represents a positive infinite quantity. }})

OpenersNotesSection(Spacer1)Details

∞ can be entered as (RefLink[\[Infinity], paclet:ref/character/Infinity]) or (Key0[esc]) inf (Key0[esc]).

In (RefLink[StandardForm, paclet:ref/StandardForm]), (RefLink[Infinity, paclet:ref/Infinity]) is printed as ∞.

(RefLink[Infinity, paclet:ref/Infinity]) is converted to (RefLink[DirectedInfinity, paclet:ref/DirectedInfinity])[1].

Certain arithmetic operations work with (RefLink[Infinity, paclet:ref/Infinity]).

(RefLink[NumberQ, paclet:ref/NumberQ])[Infinity] yields (RefLink[False, paclet:ref/False]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (12)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Use as iterator limit:

Sum[1/n^2,{n,Infinity}]

Do arithmetic with infinite quantities:

1/Infinity

OpenersExampleSection1(Spacer1)Scope  (7)

Use (Key0[esc]) inf (Key0[esc]) to enter ∞:

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 Infinity

2 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<Infinity

Find 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]

(RefLink[Cases, paclet:ref/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 (RefLink[Infinity, paclet:ref/Infinity]) to include all levels:

Cases[e,_Gamma,Infinity]

OpenersExampleSection2(Spacer1)Applications  (2)

Stirling's asymptotic approximation:

Series[n!,{n,Infinity,2}]

Setting the precision to (RefLink[Infinity, paclet:ref/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 (RefLink[SetPrecision, paclet:ref/SetPrecision]) to the whole expression to rationalize all inexact numbers:

CylindricalDecomposition[SetPrecision[y^2+2.5 x^2<1.5,Infinity],{x,y}]

OpenersExampleSection3(Spacer1)Neat Examples  (1)

Use an infinite (RefLink[WorkingPrecision, paclet:ref/WorkingPrecision]) in (RefLink[Plot, paclet:ref/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 ]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[ComplexInfinity, paclet:ref/ComplexInfinity])  ▪  (RefLink[Indeterminate, paclet:ref/Indeterminate])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Some Mathematical Functions, paclet:tutorial/SomeMathematicalFunctions])}, {▪(RefLinkPlain[Levels in Expressions, paclet:tutorial/LevelsInExpressions])}, {▪(RefLinkPlain[Indeterminate and Infinite Results, paclet:tutorial/IndeterminateAndInfiniteResults])}, {▪(RefLinkPlain[Mathematical Constants, paclet:tutorial/MathematicalConstants])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Mathematical Constants, paclet:guide/MathematicalConstants])}})}})

({{(Spacer1), ({{(Spacer1)Related Links}, {(WebLink[DocumentationFooterIconsRelatedLinks-MathWorld.png(Spacer1)MathWorld, http://mathworld.wolfram.com/Infinity.html])}, {DocumentationFooterIconsRelatedLinks-Functions.png(Spacer1)The Wolfram Functions Site}})}})