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variants of this functions
Factorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Summation > Infinite summation > Parameter-containing sums





http://functions.wolfram.com/06.01.23.0054.01









  


  










Input Form





Sum[(4 k)!/(4 k + 4 n + 2)!, {k, 0, Infinity}] == (1/2) (Sum[-(2^j/((4 n - j) (4 n + 1)!)), {j, 0, 4 n - 1}] + (2^(4 n + 2) Log[2])/(4 (4 n + 1)!)) + (1/2) (Sum[-(((-1)^j 4^(-1 + j) (1 + 40 n^2 + 18 j + 40 j^2 - 2 n (9 + 40 j)))/((-n + j) (1 - 2 n + 2 j) (1 - 4 n + 4 j) (4 n + 1)!)), {j, 0, n - 1}] + ((-4)^n (Pi/4 - Log[4]/4))/ (4 n + 1)!) /; Element[n, Integers] && n >= 1










Standard Form





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MathML Form







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</mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> - </mo> <mfrac> <msup> <mn> 2 </mn> <mi> j </mi> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -4 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 40 </cn> 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Rule Form





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Contributed by





Troy Kessler










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





2007-05-02





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