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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/06.01.23.0057.01









  


  








Input Form





Sum[(4 k)!/((4 k + 3 n + 1)! 9^k), {k, 0, Infinity}] == ((1 - Sqrt[3])^(3 n) (Sqrt[3] Log[Sqrt[3]/(Sqrt[3] - 1)]))/(4 (3 n)!) + (1/4) Sum[(Sqrt[3] (Sqrt[3] + 1)^j - (1 - Sqrt[3])^j Sqrt[3])/ ((-3 n + j) (3 n)!), {j, 0, 3 n - 1}] + ((Sqrt[3] + 1)^(3 n) (Sqrt[3] Log[(Sqrt[3] + 1)/Sqrt[3]]))/(4 (3 n)!) + (1/2) (Sum[-((3 (-8)^j (4 - 9 n + 9 j))/((1 - 3 n + 3 j) (2 - 3 n + 3 j) (3 n)!)), {j, 0, n - 1}] + ((-8)^n Pi)/((3 n (3 n - 1)!) (2 Sqrt[3]))) /; Element[n, Integers] && n >= 1








Standard Form





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MathML Form







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</apply> </apply> <apply> <power /> <cn type='integer'> 9 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <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 /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> -8 </cn> <ci> j </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </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