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Glaisher






Mathematica Notation

Traditional Notation









Constants > Glaisher > Limit representations





http://functions.wolfram.com/02.08.09.0003.01









  


  










Input Form





Glaisher == Limit[Exp[(1/12) Log[2 Pi] + EulerGamma/12 - (1/(2 Pi^2)) (2^(1 - Ceiling[1 + n Log[8, 10]]) Sum[(Log[2 (1 + j)]/(1 + j)^2) (-1)^j (Sum[Binomial[Ceiling[1 + n Log[8, 10]], k], {k, 0, j - Ceiling[1 + n Log[8, 10]]}] - 2^Ceiling[1 + n Log[8, 10]]), {j, 0, -1 + 2 Ceiling[1 + n Log[8, 10]]}])], n -> Infinity]










Standard Form





Cell[BoxData[RowBox[List["Glaisher", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "+", FractionBox["EulerGamma", "12"], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]], "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["j", "-", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]], ",", "k"]], "]"]]]], "-", SuperscriptBox["2", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], ")"]]]]]]]], ")"]]]]]], "]"]], ",", RowBox[List["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> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox[&quot;A&quot;, Function[List[], Glaisher]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mn> 8 </mn> </msub> <mo> ( </mo> <mn> 10 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8969; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mn> 8 </mn> </msub> <mo> ( </mo> <mn> 10 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8969; </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mn> 8 </mn> </msub> <mo> ( </mo> <mn> 10 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8969; </mo> </mrow> </mrow> </munderover> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mn> 8 </mn> </msub> <mo> ( </mo> <mn> 10 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8969; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;\[LeftCeiling]&quot;, RowBox[List[RowBox[List[&quot;n&quot;, &quot; &quot;, RowBox[List[SubscriptBox[&quot;log&quot;, &quot;8&quot;], &quot;(&quot;, &quot;10&quot;, &quot;)&quot;]]]], &quot;+&quot;, &quot;1&quot;]], &quot;\[RightCeiling]&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mn> 8 </mn> </msub> <mo> ( </mo> <mn> 10 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8969; </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mn> 12 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <ci> Glaisher </ci> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <exp /> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 8 </cn> </logbase> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 8 </cn> </logbase> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 8 </cn> </logbase> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </uplimit> <apply> <ci> Binomial </ci> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 8 </cn> </logbase> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 8 </cn> </logbase> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <eulergamma /> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "Glaisher", "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "+", FractionBox["EulerGamma", "12"], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]]]], FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["j", "-", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]], ",", "k"]], "]"]]]], "-", SuperscriptBox["2", RowBox[List["Ceiling", "[", RowBox[List["1", "+", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["8", ",", "10"]], "]"]]]]]], "]"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]], "2"]]]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"]]]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]










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





2007-05-02





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