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 Pi

 http://functions.wolfram.com/02.03.06.0057.01

 Input Form

 Pi^3 == (1/16) Sum[((-1)^k/1024^k) (32/(1 + 4 k)^3 + 8/(2 + 4 k)^3 + 1/(3 + 4 k)^3), {k, 0, Infinity}] + (5/2) Sum[((-1)^k/64^k) (32/(1 + 12 k)^3 - 192/(2 + 12 k)^3 + 88/(3 + 12 k)^3 - 8/(5 + 12 k)^3 + 84/(6 + 12 k)^3 - 4/(7 + 12 k)^3 + 11/(9 + 12 k)^3 - 12/(10 + 12 k)^3 + 1/(11 + 12 k)^3), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[SuperscriptBox["\[Pi]", "3"], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "16"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", "1024", ")"]], "k"]], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["5", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", "64", ")"]], "k"]], RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["192", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["88", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["84", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["11", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["12", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]]]], ")"]]]]]]]]]]]]]]

 MathML Form

 π 3 1 16 k = 0 ( - 1 ) k 1024 k ( 8 ( 4 k + 2 ) 3 + 1 ( 4 k + 3 ) 3 + 32 ( 4 k + 1 ) 3 ) + 5 2 k = 0 ( - 1 ) k 64 k ( - 192 ( 12 k + 2 ) 3 + 88 ( 12 k + 3 ) 3 - 8 ( 12 k + 5 ) 3 + 84 ( 12 k + 6 ) 3 - 4 ( 12 k + 7 ) 3 + 11 ( 12 k + 9 ) 3 - 12 ( 12 k + 10 ) 3 + 1 ( 12 k + 11 ) 3 + 32 ( 12 k + 1 ) 3 ) 3 1 16 k 0 -1 k 1024 k -1 8 4 k 2 3 -1 1 4 k 3 3 -1 32 4 k 1 3 -1 5 2 k 0 -1 k 64 k -1 -1 192 12 k 2 3 -1 88 12 k 3 3 -1 -1 8 12 k 5 3 -1 84 12 k 6 3 -1 -1 4 12 k 7 3 -1 11 12 k 9 3 -1 -1 12 12 k 10 3 -1 1 12 k 11 3 -1 32 12 k 1 3 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[Pi]", "3"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "16"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]]]], ")"]]]], SuperscriptBox["1024", "k"]]]]]], "+", RowBox[List[FractionBox["5", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["192", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["88", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["84", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["11", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["12", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]]]], ")"]]]], SuperscriptBox["64", "k"]]]]]]]]]]]]

 Contributed by

 G.Huvent (2006)

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

 2007-05-02