Factorial: Series representations (formula 06.01.06.0008)

Factorial

$Mathematica Notation$

Gamma, Beta, Erf

Factorial[n]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/06.01.06.0008.01

Input Form

n! \[Proportional] (Sqrt[2 Pi] n^(n + 1/2) (1 + 1/(12 n) + 1/(288 n^2) - 139/(51840 n^3) - 571/(2488320 n^4) + 163879/(209018880 n^5) + 5246819/(75246796800 n^6) - 534703531/(902961561600 n^7) - 4483131259/(86684309913600 n^8) + 432261921612371/ (514904800886784000 n^9) + O[1/n^10]))/E^n /; Abs[Arg[n]] < Pi && (Abs[n] -> Infinity)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!"]], "\[Proportional]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], SuperscriptBox["n", RowBox[List["n", "+", FractionBox["1", "2"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", "n"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List["12", "n"]]], "+", FractionBox["1", RowBox[List["288", SuperscriptBox["n", "2"]]]], "-", FractionBox["139", RowBox[List["51840", " ", SuperscriptBox["n", "3"]]]], "-", FractionBox["571", RowBox[List["2488320", " ", SuperscriptBox["n", "4"]]]], "+", FractionBox["163879", RowBox[List["209018880", " ", SuperscriptBox["n", "5"]]]], "+", FractionBox["5246819", RowBox[List["75246796800", " ", SuperscriptBox["n", "6"]]]], "-", FractionBox["534703531", RowBox[List["902961561600", " ", SuperscriptBox["n", "7"]]]], "-", FractionBox["4483131259", RowBox[List["86684309913600", " ", SuperscriptBox["n", "8"]]]], "+", FractionBox["432261921612371", RowBox[List["514904800886784000", " ", SuperscriptBox["n", "9"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["n", "10"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ∝ </mo> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> n </mi> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 288 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 139 </mn> <mrow> <mn> 51840 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 571 </mn> <mrow> <mn> 2488320 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 163879 </mn> <mrow> <mn> 209018880 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 5246819 </mn> <mrow> <mn> 75246796800 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 534703531 </mn> <mrow> <mn> 902961561600 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 7 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 4483131259 </mn> <mrow> <mn> 86684309913600 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 8 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 432261921612371 </mn> <mrow> <mn> 514904800886784000 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 9 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mn> 10 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <factorial /> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 139 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 51840 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 571 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2488320 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 163879 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 209018880 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5246819 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 75246796800 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 534703531 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 902961561600 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 7 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4483131259 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 86684309913600 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 432261921612371 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 514904800886784000 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 9 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> n </ci> </apply> </apply> <pi /> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> n </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

Factorial2[n]