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http://functions.wolfram.com/04.13.11.0002.01
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BernoulliB[2 n] == (2 n)! SeriesTerm[E^t/(E^t - 1) - 1/2 - 1/t,
{t, 0, 2 n - 1}] /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "n"]], ")"]], "!"]], RowBox[List["SeriesTerm", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", "t"], RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "-", "1"]]], "-", FractionBox["1", "2"], "-", FractionBox["1", "t"]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", RowBox[List[RowBox[List["2", "n"]], "-", "1"]]]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mrow> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mrow> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]], " ", RowBox[List["SeriesTerm", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", "t"], RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "-", "1"]]], "-", FractionBox["1", "2"], "-", FractionBox["1", "t"]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
