|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/02.06.09.0014.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
EulerGamma == Limit[n^2/(n + 1) - (Gamma[1/n] Gamma[n + 1] n^(1 + 1/n))/
Gamma[n + 1/n + 2], n -> Infinity]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List["EulerGamma", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["n", "2"], RowBox[List["n", "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox["1", "n"], "]"]], RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], SuperscriptBox["n", RowBox[List["1", "+", FractionBox["1", "n"]]]]]], RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "n"], "+", "2"]], "]"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mo>  </mo> <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> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> n </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <eulergamma /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "EulerGamma", "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["n", "2"], RowBox[List["n", "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox["1", "n"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", SuperscriptBox["n", RowBox[List["1", "+", FractionBox["1", "n"]]]]]], RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "n"], "+", "2"]], "]"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|