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 Pi

 http://functions.wolfram.com/02.03.06.0058.01

 Input Form

 Pi^4 == (27/164) Sum[(1/2^(12 k)) (2048/(1 + 24 k)^4 - 38912/(2 + 24 k)^4 + 81920/(3 + 24 k)^4 - 2048/(4 + 24 k)^4 - 512/(5 + 24 k)^4 - 23552/(6 + 24 k)^4 + 256/(7 + 24 k)^4 - 27648/(8 + 24 k)^4 - 10240/(9 + 24 k)^4 - 2432/(10 + 24 k)^4 - 64/(11 + 24 k)^4 - 3584/(12 + 24 k)^4 - 32/(13 + 24 k)^4 - 608/(14 + 24 k)^4 - 1280/(15 + 24 k)^4 - 1728/(16 + 24 k)^4 + 8/(17 + 24 k)^4 - 368/(18 + 24 k)^4 - 4/(19 + 24 k)^4 - 8/(20 + 24 k)^4 + 160/(21 + 24 k)^4 - 38/(22 + 24 k)^4 + 1/(23 + 24 k)^4), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[SuperscriptBox["\[Pi]", "4"], "\[Equal]", RowBox[List[FractionBox["27", "164"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["2", RowBox[List["12", " ", "k"]]], " "]]], " ", RowBox[List["(", RowBox[List[FractionBox["2048", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["38912", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["81920", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["2048", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["512", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["23552", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["256", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["27648", SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["10240", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["2432", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["64", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["3584", SuperscriptBox[RowBox[List["(", RowBox[List["12", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["13", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["608", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["1280", SuperscriptBox[RowBox[List["(", RowBox[List["15", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["1728", SuperscriptBox[RowBox[List["(", RowBox[List["16", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["17", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["368", SuperscriptBox[RowBox[List["(", RowBox[List["18", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["19", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["20", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["160", SuperscriptBox[RowBox[List["(", RowBox[List["21", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["38", SuperscriptBox[RowBox[List["(", RowBox[List["22", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["23", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]]]], ")"]]]]]]]]]]]]

 MathML Form

 π 4 27 164 k = 0 1 2 12 k ( - 38912 ( 24 k + 2 ) 4 + 81920 ( 24 k + 3 ) 4 - 2048 ( 24 k + 4 ) 4 - 512 ( 24 k + 5 ) 4 - 23552 ( 24 k + 6 ) 4 + 256 ( 24 k + 7 ) 4 - 27648 ( 24 k + 8 ) 4 - 10240 ( 24 k + 9 ) 4 - 2432 ( 24 k + 10 ) 4 - 64 ( 24 k + 11 ) 4 - 3584 ( 24 k + 12 ) 4 - 32 ( 24 k + 13 ) 4 - 608 ( 24 k + 14 ) 4 - 1280 ( 24 k + 15 ) 4 - 1728 ( 24 k + 16 ) 4 + 8 ( 24 k + 17 ) 4 - 368 ( 24 k + 18 ) 4 - 4 ( 24 k + 19 ) 4 - 8 ( 24 k + 20 ) 4 + 160 ( 24 k + 21 ) 4 - 38 ( 24 k + 22 ) 4 + 1 ( 24 k + 23 ) 4 + 2048 ( 24 k + 1 ) 4 ) 4 27 164 k 0 1 2 12 k -1 -1 38912 24 k 2 4 -1 81920 24 k 3 4 -1 -1 2048 24 k 4 4 -1 -1 512 24 k 5 4 -1 -1 23552 24 k 6 4 -1 256 24 k 7 4 -1 -1 27648 24 k 8 4 -1 -1 10240 24 k 9 4 -1 -1 2432 24 k 10 4 -1 -1 64 24 k 11 4 -1 -1 3584 24 k 12 4 -1 -1 32 24 k 13 4 -1 -1 608 24 k 14 4 -1 -1 1280 24 k 15 4 -1 -1 1728 24 k 16 4 -1 8 24 k 17 4 -1 -1 368 24 k 18 4 -1 -1 4 24 k 19 4 -1 -1 8 24 k 20 4 -1 160 24 k 21 4 -1 -1 38 24 k 22 4 -1 1 24 k 23 4 -1 2048 24 k 1 4 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[Pi]", "4"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["27", "164"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[FractionBox["2048", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["38912", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["81920", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["2048", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["512", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["23552", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["256", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["27648", SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["10240", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["2432", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["64", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["3584", SuperscriptBox[RowBox[List["(", RowBox[List["12", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["13", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["608", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["1280", SuperscriptBox[RowBox[List["(", RowBox[List["15", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["1728", SuperscriptBox[RowBox[List["(", RowBox[List["16", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["17", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["368", SuperscriptBox[RowBox[List["(", RowBox[List["18", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["19", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["20", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["160", SuperscriptBox[RowBox[List["(", RowBox[List["21", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "-", FractionBox["38", SuperscriptBox[RowBox[List["(", RowBox[List["22", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["23", "+", RowBox[List["24", " ", "k"]]]], ")"]], "4"]]]], SuperscriptBox["2", RowBox[List["12", " ", "k"]]]]]]]]]]]]

 Contributed by

 G.Huvent (2006)

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

 2007-05-02