html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Zeta

 http://functions.wolfram.com/10.01.06.0028.01

 Input Form

 Zeta[5] == (369/62651) Sum[((-1)^k/1024^k) (-(128/(1 + 4 k)^5) + 4/(3 + 4 k)^5 + 1/(4 + 4 k)^5), {k, 0, Infinity}] + (9/250604) Sum[(1/4096^k) (126976/(1 + 24 k)^5 - 6610944/(2 + 24 k)^5 + 33418240/(3 + 24 k)^5 - 12722176/(4 + 24 k)^5 - 31744/(5 + 24 k)^5 + 25829376/(6 + 24 k)^5 + 15872/(7 + 24 k)^5 + 38170624/(8 + 24 k)^5 - 4177280/(9 + 24 k)^5 - 413184/(10 + 24 k)^5 - 3968/(11 + 24 k)^5 - 6323008/(12 + 24 k)^5 - 1984/(13 + 24 k)^5 - 103296/(14 + 24 k)^5 - 522160/(15 + 24 k)^5 + 2385664/(16 + 24 k)^5 + 496/(17 + 24 k)^5 + 403584/(18 + 24 k)^5 - 248/(19 + 24 k)^5 - 49696/(20 + 24 k)^5 + 65270/(21 + 24 k)^5 - 6456/(22 + 24 k)^5 + 62/(23 + 24 k)^5 + 128125/(24 + 24 k)^5), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", "5", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["369", " "]], "62651"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", "1024", ")"]], "k"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["128", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]]]], "+", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["9", "250604"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["4096", "k"], " "]]], RowBox[List["(", RowBox[List[FractionBox["126976", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6610944", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["33418240", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["12722176", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["31744", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["25829376", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["15872", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["38170624", SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["4177280", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["413184", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["3968", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6323008", SuperscriptBox[RowBox[List["(", RowBox[List["12", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["1984", SuperscriptBox[RowBox[List["(", RowBox[List["13", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["103296", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["522160", SuperscriptBox[RowBox[List["(", RowBox[List["15", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["2385664", SuperscriptBox[RowBox[List["(", RowBox[List["16", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["496", SuperscriptBox[RowBox[List["(", RowBox[List["17", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["403584", SuperscriptBox[RowBox[List["(", RowBox[List["18", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["248", SuperscriptBox[RowBox[List["(", RowBox[List["19", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["49696", SuperscriptBox[RowBox[List["(", RowBox[List["20", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["65270", SuperscriptBox[RowBox[List["(", RowBox[List["21", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6456", SuperscriptBox[RowBox[List["(", RowBox[List["22", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["62", SuperscriptBox[RowBox[List["(", RowBox[List["23", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["128125", SuperscriptBox[RowBox[List["(", RowBox[List["24", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]]]], ")"]]]]]]]]]]]]]]

 MathML Form

 ζ ( 5 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["5", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] 369 62651 k = 0 ( - 1 ) k 1024 k ( 4 ( 4 k + 3 ) 5 + 1 ( 4 k + 4 ) 5 - 128 ( 4 k + 1 ) 5 ) + 9 250604 k = 0 1 4096 k ( - 6610944 ( 24 k + 2 ) 5 + 33418240 ( 24 k + 3 ) 5 - 12722176 ( 24 k + 4 ) 5 - 31744 ( 24 k + 5 ) 5 + 25829376 ( 24 k + 6 ) 5 + 15872 ( 24 k + 7 ) 5 + 38170624 ( 24 k + 8 ) 5 - 4177280 ( 24 k + 9 ) 5 - 413184 ( 24 k + 10 ) 5 - 3968 ( 24 k + 11 ) 5 - 6323008 ( 24 k + 12 ) 5 - 1984 ( 24 k + 13 ) 5 - 103296 ( 24 k + 14 ) 5 - 522160 ( 24 k + 15 ) 5 + 2385664 ( 24 k + 16 ) 5 + 496 ( 24 k + 17 ) 5 + 403584 ( 24 k + 18 ) 5 - 248 ( 24 k + 19 ) 5 - 49696 ( 24 k + 20 ) 5 + 65270 ( 24 k + 21 ) 5 - 6456 ( 24 k + 22 ) 5 + 62 ( 24 k + 23 ) 5 + 128125 ( 24 k + 24 ) 5 + 126976 ( 24 k + 1 ) 5 ) Zeta 5 369 62651 k 0 -1 k 1024 k -1 4 4 k 3 5 -1 1 4 k 4 5 -1 -1 128 4 k 1 5 -1 9 250604 k 0 1 4096 k -1 -1 6610944 24 k 2 5 -1 33418240 24 k 3 5 -1 -1 12722176 24 k 4 5 -1 -1 31744 24 k 5 5 -1 25829376 24 k 6 5 -1 15872 24 k 7 5 -1 38170624 24 k 8 5 -1 -1 4177280 24 k 9 5 -1 -1 413184 24 k 10 5 -1 -1 3968 24 k 11 5 -1 -1 6323008 24 k 12 5 -1 -1 1984 24 k 13 5 -1 -1 103296 24 k 14 5 -1 -1 522160 24 k 15 5 -1 2385664 24 k 16 5 -1 496 24 k 17 5 -1 403584 24 k 18 5 -1 -1 248 24 k 19 5 -1 -1 49696 24 k 20 5 -1 65270 24 k 21 5 -1 -1 6456 24 k 22 5 -1 62 24 k 23 5 -1 128125 24 k 24 5 -1 126976 24 k 1 5 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "5", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["369", " ", 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["-", FractionBox["128", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]]]], "+", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["4", " ", "k"]]]], ")"]], "5"]]]], ")"]]]], SuperscriptBox["1024", "k"]]]]]], "62651"], "+", FractionBox[RowBox[List["9", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[FractionBox["126976", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6610944", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["33418240", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["12722176", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["31744", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["25829376", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["15872", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["38170624", SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["4177280", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["413184", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["3968", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6323008", SuperscriptBox[RowBox[List["(", RowBox[List["12", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["1984", SuperscriptBox[RowBox[List["(", RowBox[List["13", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["103296", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["522160", SuperscriptBox[RowBox[List["(", RowBox[List["15", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["2385664", SuperscriptBox[RowBox[List["(", RowBox[List["16", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["496", SuperscriptBox[RowBox[List["(", RowBox[List["17", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["403584", SuperscriptBox[RowBox[List["(", RowBox[List["18", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["248", SuperscriptBox[RowBox[List["(", RowBox[List["19", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["49696", SuperscriptBox[RowBox[List["(", RowBox[List["20", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["65270", SuperscriptBox[RowBox[List["(", RowBox[List["21", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "-", FractionBox["6456", SuperscriptBox[RowBox[List["(", RowBox[List["22", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["62", SuperscriptBox[RowBox[List["(", RowBox[List["23", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]], "+", FractionBox["128125", SuperscriptBox[RowBox[List["(", RowBox[List["24", "+", RowBox[List["24", " ", "k"]]]], ")"]], "5"]]]], SuperscriptBox["4096", "k"]]]]]], "250604"]]]]]]]

 Contributed by

 G.Huvent (2006)

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

 2007-05-02