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http://functions.wolfram.com/04.13.25.0001.01
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Limit[Abs[BernoulliB[2 n]] ((Pi E)/n)^(2 n + 1/2), n -> Infinity] ==
4 Pi Sqrt[E]
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Cell[BoxData[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Pi]", " ", "\[ExponentialE]"]], "n"], ")"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", FractionBox["1", "2"]]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List["4", " ", "\[Pi]", " ", SqrtBox["\[ExponentialE]"]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅇ </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mi> ⅇ </mi> </msqrt> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <abs /> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <exponentiale /> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <power /> <exponentiale /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n_"]], "]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Pi]", " ", "\[ExponentialE]"]], "n_"], ")"]], RowBox[List[RowBox[List["2", " ", "n_"]], "+", FractionBox["1", "2"]]]]]], ",", RowBox[List["n_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["4", " ", "\[Pi]", " ", SqrtBox["\[ExponentialE]"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
