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Pi






Mathematica Notation

Traditional Notation









Constants > Pi > Series representations > Generalized power series > Expansions for Pi4





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





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MathML Form







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</math>










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





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