Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Factorial




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.01.23.0034.01









  


  








Input Form





Sum[((-1)^k (k + n)!)/(k! p^k), {k, 0, Infinity}] == n! (p/(p + 1))^(n + 1) /; Element[n, Integers] && n > 0 && Element[p, Integers] && p > 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", SuperscriptBox["p", "k"]]]]]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], SuperscriptBox[RowBox[List["(", FractionBox["p", RowBox[List["p", "+", "1"]]], ")"]], RowBox[List["n", "+", "1"]]]]]]], " ", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "1"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> p </mi> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </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> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <ci> p </ci> <apply> <power /> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> p </ci> <ci> &#8469; </ci> </apply> <apply> <gt /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k_"], " ", RowBox[List[RowBox[List["(", RowBox[List["k_", "+", "n_"]], ")"]], "!"]]]], RowBox[List[RowBox[List["k_", "!"]], " ", SuperscriptBox["p_", "k_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["p", RowBox[List["p", "+", "1"]]], ")"]], RowBox[List["n", "+", "1"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "1"]]]]]]]]]]










Contributed by





Troy Kessler


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





2007-05-02