(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 63505, 2180] NotebookOptionsPosition[ 54189, 1855] NotebookOutlinePosition[ 55523, 1899] CellTagsIndexPosition[ 55227, 1889] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solutions: ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" 6 Lab 0", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Subtitle", CellChangeTimes->{{3.396207261561866*^9, 3.396207292942472*^9}, { 3.3963074729456472`*^9, 3.396307474846087*^9}, {3.396449070635025*^9, 3.396449107599749*^9}}], Cell[TextData[{ StyleBox["Problem 1", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962073124090643`*^9, 3.3962073263052273`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Binomial"}]], "Input", CellChangeTimes->{{3.396207384278738*^9, 3.39620738819481*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Binomial\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"n\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"m\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) gives the binomial coefficient \ \\!\\(\\*RowBox[{\\\"(\\\", GridBox[{{\\\"n\\\"}, {\\\"m\\\"}}], \ \\\")\\\"}]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Binomial"]}]], "Print", "PrintUsage", CellChangeTimes->{3.396207394031246*^9}, CellTags->"Info3396182190-1141342"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Binomial", "[", RowBox[{"81", ",", "14"}], "]"}]], "Input", CellChangeTimes->{{3.396207431736721*^9, 3.396207459803537*^9}}], Cell[BoxData["1823288518168200"], "Output", CellChangeTimes->{3.3962074623110456`*^9}] }, Open ]], Cell["Compare that with ", "Text", CellChangeTimes->{{3.3962075146225767`*^9, 3.396207536649354*^9}, { 3.396449326605142*^9, 3.3964493321768293`*^9}, {3.396449371022914*^9, 3.396449378942013*^9}}], Cell[CellGroupData[{ Cell[BoxData[ FractionBox[ RowBox[{"81", "!"}], RowBox[{ RowBox[{"14", "!"}], RowBox[{ RowBox[{"(", RowBox[{"81", "-", "14"}], ")"}], "!"}]}]]], "Input", CellChangeTimes->{{3.396207478480229*^9, 3.396207503100725*^9}}], Cell[BoxData["1823288518168200"], "Output", CellChangeTimes->{3.3962075057342863`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "n"], RowBox[{"Binomial", "[", RowBox[{"n", ",", "k"}], "]"}]}]], "Input", CellChangeTimes->{{3.3962075495742083`*^9, 3.396207578870625*^9}}], Cell[BoxData[ SuperscriptBox["2", "n"]], "Output", CellChangeTimes->{3.3962075835502043`*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396207609353353*^9}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 2", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396207621654594*^9, 3.396207625681943*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "k"], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "+", "1"}], ")"}], RowBox[{"-", "2"}]]}]}]], "Input"], Cell[BoxData["Catalan"], "Output", CellChangeTimes->{3.396207677846676*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Catalan"}]], "Input", CellChangeTimes->{{3.396207700488357*^9, 3.396207703837158*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"Catalan is Catalan\[CloseCurlyQuote]s constant, with \ numerical value \\!\\(\\*RowBox[{\\\"\[TildeEqual]\\\", \ \\\"0.915966\\\"}]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Catalan"]}]], "Print", "PrintUsage", CellChangeTimes->{3.3962077056844587`*^9}, CellTags->"Info3396182505-3705663"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"N", "[", "Catalan", "]"}], "\[IndentingNewLine]", RowBox[{"Catalan", "//", "N"}]}], "Input", CellChangeTimes->{{3.396208303973136*^9, 3.396208321532627*^9}}], Cell[BoxData["0.915965594177219`"], "Output", CellChangeTimes->{3.396208325928323*^9, 3.396208400431471*^9}] }, Open ]], Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Catalan", ",", "50"}], "]"}]], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Lab Number 0", "Title", CellChangeTimes->{{3.442167916901147*^9, 3.442167924626935*^9}}], Cell["Evaluated by Jeremy Silver", "Subtitle", CellChangeTimes->{{3.442167935303179*^9, 3.4421679405381517`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DateString", "[", "]"}]], "Input", CellChangeTimes->{{3.44216864881985*^9, 3.4421686529394913`*^9}}], Cell[BoxData["\<\"Wed 28 Jan 2009 14:16:19\"\>"], "Output", CellChangeTimes->{ 3.442168653481265*^9, 3.442169418907052*^9, {3.442169767545204*^9, 3.4421697793978786`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Problem 1", "Section", CellChangeTimes->{{3.442167944619071*^9, 3.442167946658868*^9}, { 3.4421680427415447`*^9, 3.442168042972665*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Binomial"}]], "Input", CellChangeTimes->{{3.4421679505805807`*^9, 3.442167956948021*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Binomial\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"n\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"m\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) gives the binomial coefficient \ \\!\\(\\*RowBox[{\\\"(\\\", GridBox[{{\\\"n\\\"}, {\\\"m\\\"}}], \ \\\")\\\"}]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Binomial"]}]], "Print", "PrintUsage", CellChangeTimes->{3.4421697797600117`*^9}, CellTags->"Info3442140979-7902732"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Binomial", "[", RowBox[{"81", ",", "14"}], "]"}]], "Input", CellChangeTimes->{{3.442168027454694*^9, 3.4421680370923643`*^9}}], Cell[BoxData["1823288518168200"], "Output", CellChangeTimes->{ 3.442168037712543*^9, 3.4421694193389587`*^9, {3.4421697678142242`*^9, 3.4421697798628683`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "n"], RowBox[{"Binomial", "[", RowBox[{"n", ",", "k"}], "]"}]}]], "Input", CellChangeTimes->{{3.442169502623252*^9, 3.442169519727034*^9}}], Cell[BoxData[ SuperscriptBox["2", "n"]], "Output", CellChangeTimes->{ 3.442169520805629*^9, {3.4421697678430233`*^9, 3.442169779919507*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "Section", CellChangeTimes->{{3.442168046806662*^9, 3.442168048716337*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "k"], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "+", "1"}], ")"}], RowBox[{"-", "2"}]]}]}]], "Input", CellChangeTimes->{{3.4421681058800917`*^9, 3.442168150998722*^9}, { 3.4421683500670967`*^9, 3.4421683532640743`*^9}}], Cell[BoxData["Catalan"], "Output", CellChangeTimes->{ 3.442168152599238*^9, 3.4421694194256973`*^9, {3.44216976787617*^9, 3.442169779965288*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Catalan"}]], "Input", CellChangeTimes->{{3.442168158022078*^9, 3.442168162593289*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"Catalan is Catalan\[CloseCurlyQuote]s constant, with \ numerical value \\!\\(\\*RowBox[{\\\"\[TildeEqual]\\\", \ \\\"0.915966\\\"}]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Catalan"]}]], "Print", "PrintUsage", CellChangeTimes->{3.442169780093092*^9}, CellTags->"Info3442140979-6828370"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Catalan", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.442168167005012*^9, 3.442168175197975*^9}}], Cell[BoxData["0.\ 9159655941772190150546035149323841107741493742816719318884420313396`50."], \ "Output", CellChangeTimes->{ 3.442168175759944*^9, 3.44216941968896*^9, {3.442169768097492*^9, 3.442169780229353*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.4421682804242268`*^9, 3.442168335296666*^9}, { 3.442169545650407*^9, 3.442169580584525*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Catalan Biography\"\>", True-> StyleBox["\<\"Catalan Biography\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan"], Annotation[#, "http://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan", "Hyperlink"]& ]], "Output", CellChangeTimes->{ 3.442168331212907*^9, 3.4421694199143963`*^9, {3.4421695552926702`*^9, 3.4421695819046583`*^9}, {3.442169768127359*^9, 3.4421697802593117`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem 3", "Section", CellChangeTimes->{{3.442168361518297*^9, 3.442168363441595*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "k"], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "+", "1"}], ")"}], RowBox[{"-", "1"}]]}]}]], "Input", CellChangeTimes->{{3.4421683668090277`*^9, 3.44216836824118*^9}}], Cell[BoxData[ FractionBox["\[Pi]", "4"]], "Output", CellChangeTimes->{ 3.442168369158985*^9, 3.442169420004148*^9, 3.442169658834399*^9, { 3.442169768184971*^9, 3.442169780324565*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"%", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.442168377462085*^9, 3.442168380776886*^9}, { 3.442169654594645*^9, 3.442169655081523*^9}}], Cell[BoxData["0.\ 7853981633974483096156608458198757210492923498437764552462368522683`50."], \ "Output", CellChangeTimes->{ 3.442168381215287*^9, 3.4421694200394583`*^9, 3.442169658913199*^9, { 3.44216976823123*^9, 3.442169780363446*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "Section", CellChangeTimes->{{3.4421685919574337`*^9, 3.442168594250696*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Grid", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Text", "[", SuperscriptBox["\[Pi]", "\[Pi]"], "]"}], ",", RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[Pi]", "\[Pi]"], ",", "20"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Text", "[", SuperscriptBox["\[ExponentialE]", "\[Pi]"], "]"}], ",", RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "\[Pi]"], ",", "20"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Text", "[", SuperscriptBox["\[Pi]", "\[ExponentialE]"], "]"}], ",", RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[Pi]", "\[ExponentialE]"], ",", "20"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Text", "[", SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"], "]"}], ",", RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"], ",", "20"}], "]"}]}], "}"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.4421696745957003`*^9, 3.442169682867361*^9}}], Cell[BoxData[ TagBox[GridBox[{ { InterpretationBox[Cell[BoxData[ SuperscriptBox["\[Pi]", "\[Pi]"]], "Text", "TR"], Text[Pi^Pi]], "36.46215960720791177099082603555`20."}, { InterpretationBox[Cell[BoxData[ SuperscriptBox["\[ExponentialE]", "\[Pi]"]], "Text", "TR"], Text[E^Pi]], "23.14069263277926900572908637507`20."}, { InterpretationBox[Cell[BoxData[ SuperscriptBox["\[Pi]", "\[ExponentialE]"]], "Text", "TR"], Text[Pi^E]], "22.45915771836104547342715213835`20."}, { InterpretationBox[Cell[BoxData[ SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"]], "Text", "TR"], Text[E^E]], "15.1542622414792641897604302336`20."} }, ColumnsEqual->False, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, RowsEqual->False], "Grid"]], "Output", CellChangeTimes->{{3.442168441627411*^9, 3.442168453842144*^9}, 3.442168514985756*^9, 3.442169420093574*^9, 3.442169683359922*^9, { 3.442169768281013*^9, 3.4421697804156446`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["\[Pi]", "\[Pi]"], ">", SuperscriptBox["\[ExponentialE]", "\[Pi]"], ">", SuperscriptBox["\[Pi]", "\[ExponentialE]"], ">", SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"]}]], "Input", CellChangeTimes->{{3.442168548109643*^9, 3.442168561885696*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{ 3.442168562862011*^9, 3.442169420139862*^9, {3.442169768331256*^9, 3.442169780510995*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem 5", "Section", CellChangeTimes->{{3.4421685793256474`*^9, 3.442168596787973*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], SuperscriptBox["x", "k"]}]], "Input", CellChangeTimes->{{3.442168602780963*^9, 3.442168621586059*^9}, { 3.442168676471086*^9, 3.442168752237648*^9}}], Cell[BoxData[ FractionBox["1", RowBox[{"1", "-", "x"}]]], "Output", CellChangeTimes->{ 3.442168670947069*^9, {3.44216873197087*^9, 3.442168753168692*^9}, 3.442169420189795*^9, {3.442169768380315*^9, 3.442169780562467*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "i"}], "\[Infinity]"], FractionBox["1", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "-", "1"}], ")"}], "4"]]}]], "Input", CellChangeTimes->{{3.4421687582342777`*^9, 3.442168778283133*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "96"], " ", RowBox[{"PolyGamma", "[", RowBox[{"3", ",", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "+", "i"}]}], "]"}]}]], "Output", CellChangeTimes->{ 3.442168779068447*^9, 3.442169420254922*^9, {3.4421697684418707`*^9, 3.4421697806273937`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "\[Infinity]"], FractionBox["1", RowBox[{ SuperscriptBox["2", "k"], SuperscriptBox["k", "2"]}]]}]], "Input", CellChangeTimes->{{3.4421687873642263`*^9, 3.44216881216639*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[Pi]", "2"], "-", RowBox[{"6", " ", SuperscriptBox[ RowBox[{"Log", "[", "2", "]"}], "2"]}]}], ")"}]}]], "Output", CellChangeTimes->{ 3.442168813033929*^9, 3.4421694203023977`*^9, {3.442169768483582*^9, 3.442169780673991*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "1"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"k", "+", "1"}]], RowBox[{ RowBox[{"2", "k"}], "-", "1"}]]}], ")"}]}]], "Input", CellChangeTimes->{{3.4421688180274277`*^9, 3.442168878975692*^9}}], Cell[BoxData[ SqrtBox["2"]], "Output", CellChangeTimes->{{3.442168868735236*^9, 3.44216887940796*^9}, 3.442169420578948*^9, {3.442169768532712*^9, 3.4421697809440727`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"x", RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "1"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["k", "2"], SuperscriptBox["\[Pi]", "2"]}]]}], ")"}]}]}]], "Input", CellChangeTimes->{{3.4421688867179613`*^9, 3.442168921352439*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SqrtBox[ SuperscriptBox["x", "2"]], " ", RowBox[{"Sin", "[", SqrtBox[ SuperscriptBox["x", "2"]], "]"}]}], "x"]], "Output", CellChangeTimes->{ 3.442168921942554*^9, 3.442169420630329*^9, {3.442169768580861*^9, 3.4421697810039043`*^9}}] }, Open ]], Cell["\<\ Use FullSimplify to get rid of the radicals. Automatic in v7.\ \>", "Text", CellChangeTimes->{{3.442346622333643*^9, 3.442346641731717*^9}}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["Problem 6", "Section", CellChangeTimes->{{3.442168927316841*^9, 3.442168929752522*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "3"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], "\[IndentingNewLine]", RowBox[{"TraditionalForm", "[", "%", "]"}], "\[IndentingNewLine]", RowBox[{"TeXForm", "[", "%", "]"}]}], "Input", CellChangeTimes->{{3.442168933075287*^9, 3.442169047770192*^9}}], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"1", "+", RowBox[{"2", " ", "x"}]}], SqrtBox["3"]], "]"}], SqrtBox["3"]], "-", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], "]"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"]}], "]"}]}]}]], "Output", CellChangeTimes->{ 3.4421689751311913`*^9, 3.4421690055255747`*^9, 3.4421690487516603`*^9, 3.442169420690213*^9, {3.4421697686822863`*^9, 3.442169781063527*^9}}], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", FractionBox[ RowBox[{ RowBox[{"2", " ", "x"}], "+", "1"}], SqrtBox["3"]], ")"}], SqrtBox["3"]], "-", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"log", "(", RowBox[{"x", "-", "1"}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", "x", "+", "1"}], ")"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{ 3.4421689751311913`*^9, 3.4421690055255747`*^9, 3.4421690487516603`*^9, 3.442169420690213*^9, {3.4421697686822863`*^9, 3.4421697810784817`*^9}}], Cell["\<\ \\frac{\\tan ^{-1}\\left(\\frac{2 x+1}{\\sqrt{3}}\\right)}{\\sqrt{3}}-\\frac{1}{3} \\log (x-1)+\\frac{1}{6} \\log \\left(x^2+x+1\\right)\ \>", "Output", CellChangeTimes->{ 3.4421689751311913`*^9, 3.4421690055255747`*^9, 3.4421690487516603`*^9, 3.442169420690213*^9, {3.4421697686822863`*^9, 3.442169781107983*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "5"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], "//", "TraditionalForm"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "7"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], "//", "TraditionalForm"}]}], "Input", CellChangeTimes->{{3.442169060236006*^9, 3.442169100236775*^9}}], Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", "20"], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", SqrtBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"5", "+", SqrtBox["5"]}], ")"}]}]], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", FractionBox[ RowBox[{ RowBox[{"4", " ", "x"}], "-", SqrtBox["5"], "+", "1"}], SqrtBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"5", "+", SqrtBox["5"]}], ")"}]}]]], ")"}]}], "+", RowBox[{"2", " ", SqrtBox[ RowBox[{"10", "-", RowBox[{"2", " ", SqrtBox["5"]}]}]], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", FractionBox[ RowBox[{ RowBox[{"4", " ", "x"}], "+", SqrtBox["5"], "+", "1"}], SqrtBox[ RowBox[{"10", "-", RowBox[{"2", " ", SqrtBox["5"]}]}]]], ")"}]}], "-", RowBox[{"4", " ", RowBox[{"log", "(", RowBox[{"x", "-", "1"}], ")"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}], " ", "x"}], "+", "1"}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}], " ", "x"}], "+", "1"}], ")"}]}]}], ")"}]}], TraditionalForm]], "Output", CellChangeTimes->{{3.44216909126509*^9, 3.442169101179826*^9}, 3.442169421042861*^9, {3.4421697687800713`*^9, 3.442169781410544*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox["2", "7"], " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "7"], ")"}], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"(", RowBox[{"x", "+", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}]}], ")"}], " ", RowBox[{"csc", "(", FractionBox["\[Pi]", "7"], ")"}]}], ")"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{"x", "-", "1"}], ")"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}], " ", "x"}], "+", "1"}], ")"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", " ", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}], " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}], " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"sec", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}], " ", RowBox[{"(", RowBox[{"x", "-", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], ")"}]}], ")"}], " ", RowBox[{"cos", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"sec", "(", FractionBox["\[Pi]", "14"], ")"}], " ", RowBox[{"(", RowBox[{"x", "+", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}]}], ")"}]}], ")"}], " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "14"], ")"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{{3.44216909126509*^9, 3.442169101179826*^9}, 3.442169421042861*^9, {3.4421697687800713`*^9, 3.4421697814278584`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem 7", "Section", CellChangeTimes->{{3.442169118810521*^9, 3.442169121066752*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Table"}]], "Input", CellChangeTimes->{{3.442169123940166*^9, 3.4421691251300488`*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates a list of \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]]\\) copies of \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\). \\n\ \\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"i\\\", \\\"TI\ \\\"], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates \ a list of the values of \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\) when \ \\!\\(\\*StyleBox[\\\"i\\\", \\\"TI\\\"]\\) runs from 1 to \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) starts \ with \\!\\(\\*RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\"=\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]]}]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]], \\\",\\\", StyleBox[\\\"di\\\", \ \\\"TI\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) uses steps \ \\!\\(\\*StyleBox[\\\"di\\\", \\\"TI\\\"]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\ \", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \ \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) \ uses the successive values \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\ \"], StyleBox[\\\"1\\\", \\\"TR\\\"]]\\), \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\ \"TR\\\"]]\\), \\!\\(\\*StyleBox[\\\"\[Ellipsis]\\\", \ \\\"TR\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\ \", RowBox[{StyleBox[\\\"j\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"j\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"j\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"]\\\"}]\\) gives a nested list. The list \ associated with \\!\\(\\*StyleBox[\\\"i\\\", \\\"TI\\\"]\\) is outermost. \ \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Table"]}]], "Print", "PrintUsage", CellChangeTimes->{3.4421697816799707`*^9}, CellTags->"Info3442140981-9562016"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"triangularnumbers", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], "k"}], ",", RowBox[{"{", RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.442169127954875*^9, 3.442169228972198*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "3", ",", "6", ",", "10", ",", "15", ",", "21", ",", "28", ",", "36", ",", "45", ",", "55"}], "}"}]], "Output", CellChangeTimes->{{3.442169184135948*^9, 3.4421692332620573`*^9}, 3.4421694214709806`*^9, {3.4421697691144133`*^9, 3.4421697818127117`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], SuperscriptBox["k", "3"]}], ",", RowBox[{"{", RowBox[{"n", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4421692494073143`*^9, 3.442169295741006*^9}, 3.442169761621461*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "9", ",", "36", ",", "100", ",", "225", ",", "441", ",", "784", ",", "1296", ",", "2025", ",", "3025"}], "}"}]], "Output", CellChangeTimes->{ 3.442169296438264*^9, 3.442169421522307*^9, {3.442169762215157*^9, 3.442169781862537*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ SuperscriptBox["triangularnumbers", "2"]], "Input", CellChangeTimes->{{3.442169299250951*^9, 3.4421693058675737`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "9", ",", "36", ",", "100", ",", "225", ",", "441", ",", "784", ",", "1296", ",", "2025", ",", "3025"}], "}"}]], "Output", CellChangeTimes->{ 3.44216930631912*^9, 3.4421694215727262`*^9, {3.44216976921441*^9, 3.442169781912842*^9}}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], SuperscriptBox["k", "3"]}], "==", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], "k"}], ")"}], "2"]}]], "Input", CellChangeTimes->{{3.4421693207012997`*^9, 3.4421693411343927`*^9}}], Cell[CellGroupData[{ Cell[BoxData["True"], "Input"], Cell[BoxData["0.\ 9159655941772190150546035149323841107741493742816719318884420313396`50."], \ "Output", CellChangeTimes->{3.396208412556967*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.396208076291299*^9, 3.396208121032187*^9}, 3.396208244268134*^9}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"CatalanBiography\"\>", True-> StyleBox["\<\"CatalanBiography\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Eugene_Charles_Catalan"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Eugene_Charles_Catalan"], Annotation[#, "http://en.wikipedia.org/wiki/Eugene_Charles_Catalan", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.396208133446925*^9, 3.3962082589848757`*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396208453554154*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 3", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396208463035427*^9, 3.396208467663378*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "k"], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "+", "1"}], ")"}], RowBox[{"-", "1"}]]}]}]], "Input"], Cell[BoxData[ FractionBox["\[Pi]", "4"]], "Output", CellChangeTimes->{3.396208513539143*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"%", ",", "50"}], "]"}]], "Input", CellChangeTimes->{{3.396208527501441*^9, 3.396208532227787*^9}}], Cell[BoxData["0.\ 7853981633974483096156608458198757210492923498437764552462368522683`50."], \ "Output", CellChangeTimes->{3.396208535159504*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396208569536446*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 4", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396208617694413*^9, 3.396208620457095*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[Pi]", "\[Pi]"], ",", "20"}], "]"}]], "Input", CellChangeTimes->{3.3962087118858423`*^9}], Cell[BoxData["36.46215960720791177099082603555`20."], "Output", CellChangeTimes->{3.3962087153513527`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "\[Pi]"], ",", "20"}], "]"}]], "Input"], Cell[BoxData["23.14069263277926900572908637507`20."], "Output", CellChangeTimes->{3.396208717607699*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[Pi]", "\[ExponentialE]"], ",", "20"}], "]"}]], "Input"], Cell[BoxData["22.45915771836104547342715213835`20."], "Output", CellChangeTimes->{3.396208719960393*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"], ",", "20"}], "]"}]], "Input"], Cell[BoxData["15.1542622414792641897604302336`20."], "Output", CellChangeTimes->{3.396208722309794*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"N", "[", SuperscriptBox["\[Pi]", "\[Pi]"], "]"}], ">", RowBox[{"N", "[", SuperscriptBox["\[ExponentialE]", "\[Pi]"], "]"}], ">", RowBox[{"N", "[", SuperscriptBox["\[Pi]", "\[ExponentialE]"], "]"}], ">", RowBox[{"N", "[", SuperscriptBox["\[ExponentialE]", "\[ExponentialE]"], "]"}]}]], "Input"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396208725838264*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396208759402788*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 5", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962087742059317`*^9, 3.396208780163588*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]"], SuperscriptBox["x", "k"]}]], "Input"], Cell[BoxData[ FractionBox["1", RowBox[{"1", "-", "x"}]]], "Output", CellChangeTimes->{3.396208876511297*^9}] }, Open ]], Cell[TextData[{ "Note that ", StyleBox["Mathematica", FontSlant->"Italic"], " did not report the convergence condition." }], "Text", CellChangeTimes->{{3.396208899095619*^9, 3.396208929106565*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "\[Infinity]"], FractionBox["1", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "-", "1"}], ")"}], "4"]]}]], "Input"], Cell[BoxData[ FractionBox[ SuperscriptBox["\[Pi]", "4"], "96"]], "Output", CellChangeTimes->{3.396208950402382*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "\[Infinity]"], FractionBox["1", RowBox[{ SuperscriptBox["2", "k"], SuperscriptBox["k", "2"]}]]}]], "Input"], Cell[BoxData[ RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[Pi]", "2"], "-", RowBox[{"6", " ", SuperscriptBox[ RowBox[{"Log", "[", "2", "]"}], "2"]}]}], ")"}]}]], "Output", CellChangeTimes->{3.396208967326437*^9}] }, Open ]], Cell[TextData[{ "In the remaining exercises it is important to remember that ", StyleBox["Mathematica", FontSlant->"Italic"], " reserves curly brackets ", StyleBox["{ }", "Input"], " to identify ", StyleBox["lists", FontSlant->"Italic"], ":" }], "Text", CellChangeTimes->{{3.396449801373724*^9, 3.396449871847164*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "1"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"k", "+", "1"}]], RowBox[{ RowBox[{"2", "k"}], "-", "1"}]]}], ")"}]}]], "Input"], Cell[BoxData[ SqrtBox["2"]], "Output", CellChangeTimes->{3.3962090220405912`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"x", RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "1"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["k", "2"], SuperscriptBox["\[Pi]", "2"]}]]}], ")"}]}]}]], "Input"], Cell[BoxData[ FractionBox[ RowBox[{ SqrtBox[ SuperscriptBox["x", "2"]], " ", RowBox[{"Sin", "[", SqrtBox[ SuperscriptBox["x", "2"]], "]"}]}], "x"]], "Output", CellChangeTimes->{3.396209044818056*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"%", ",", RowBox[{"x", ">", "0"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Sin", "[", "x", "]"}]], "Output", CellChangeTimes->{3.3962090949099693`*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396209519015764*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 6", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.39620953164935*^9, 3.396209536519012*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "3"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"1", "+", RowBox[{"2", " ", "x"}]}], SqrtBox["3"]], "]"}], SqrtBox["3"]], "-", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], "]"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"]}], "]"}]}]}]], "Output", CellChangeTimes->{3.396209581622253*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TraditionalForm", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.396209594998889*^9, 3.396209603308201*^9}}], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", FractionBox[ RowBox[{ RowBox[{"2", " ", "x"}], "+", "1"}], SqrtBox["3"]], ")"}], SqrtBox["3"]], "-", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"log", "(", RowBox[{"x", "-", "1"}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", "x", "+", "1"}], ")"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{3.3962096053494453`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "5"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ FractionBox["1", "20"], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", SqrtBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"5", "+", SqrtBox["5"]}], ")"}]}]], " ", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"1", "-", SqrtBox["5"], "+", RowBox[{"4", " ", "x"}]}], SqrtBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"5", "+", SqrtBox["5"]}], ")"}]}]]], "]"}]}], "+", RowBox[{"2", " ", SqrtBox[ RowBox[{"10", "-", RowBox[{"2", " ", SqrtBox["5"]}]}]], " ", RowBox[{"ArcTan", "[", FractionBox[ RowBox[{"1", "+", SqrtBox["5"], "+", RowBox[{"4", " ", "x"}]}], SqrtBox[ RowBox[{"10", "-", RowBox[{"2", " ", SqrtBox["5"]}]}]]], "]"}]}], "-", RowBox[{"4", " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}], " ", RowBox[{"Log", "[", RowBox[{"1", "-", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["5"]}], ")"}], " ", "x"}], "+", SuperscriptBox["x", "2"]}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}], " ", RowBox[{"Log", "[", RowBox[{"1", "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SqrtBox["5"]}], ")"}], " ", "x"}], "+", SuperscriptBox["x", "2"]}], "]"}]}]}], ")"}]}]], "Output", CellChangeTimes->{3.3964501603793907`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "7"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["2", "7"], " ", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"Sec", "[", FractionBox["\[Pi]", "14"], "]"}], " ", RowBox[{"(", RowBox[{"x", "+", RowBox[{"Sin", "[", FractionBox["\[Pi]", "14"], "]"}]}], ")"}]}], "]"}], " ", RowBox[{"Cos", "[", FractionBox["\[Pi]", "14"], "]"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"Sec", "[", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], "]"}], " ", RowBox[{"(", RowBox[{"x", "-", RowBox[{"Sin", "[", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], "]"}]}], ")"}]}], "]"}], " ", RowBox[{"Cos", "[", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], "]"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"Log", "[", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], "]"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"Cos", "[", FractionBox["\[Pi]", "7"], "]"}], " ", RowBox[{"Log", "[", RowBox[{"1", "+", SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", "x", " ", RowBox[{"Cos", "[", FractionBox["\[Pi]", "7"], "]"}]}]}], "]"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", "x", " ", RowBox[{"Sin", "[", FractionBox["\[Pi]", "14"], "]"}]}]}], "]"}], " ", RowBox[{"Sin", "[", FractionBox["\[Pi]", "14"], "]"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"(", RowBox[{"x", "+", RowBox[{"Cos", "[", FractionBox["\[Pi]", "7"], "]"}]}], ")"}], " ", RowBox[{"Csc", "[", FractionBox["\[Pi]", "7"], "]"}]}], "]"}], " ", RowBox[{"Sin", "[", FractionBox["\[Pi]", "7"], "]"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", SuperscriptBox["x", "2"], "-", RowBox[{"2", " ", "x", " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], "]"}]}]}], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], "]"}]}]}]], "Output", CellChangeTimes->{3.396450204689108*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{"1", "-", SuperscriptBox["x", "7"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], "//", "TraditionalForm"}]], "Input",\ CellChangeTimes->{{3.396209722455291*^9, 3.396209725711877*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox["2", "7"], " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "7"], ")"}], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"(", RowBox[{"x", "+", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}]}], ")"}], " ", RowBox[{"csc", "(", FractionBox["\[Pi]", "7"], ")"}]}], ")"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{"x", "-", "1"}], ")"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "7"], ")"}], " ", "x"}], "+", "1"}], ")"}]}], "-", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", " ", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}], " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], "+", RowBox[{ FractionBox["1", "7"], " ", RowBox[{"log", "(", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"2", " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}], " ", "x"}], "+", "1"}], ")"}], " ", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"sec", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}], " ", RowBox[{"(", RowBox[{"x", "-", RowBox[{"sin", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], ")"}]}], ")"}], " ", RowBox[{"cos", "(", FractionBox[ RowBox[{"3", " ", "\[Pi]"}], "14"], ")"}]}], "+", RowBox[{ FractionBox["2", "7"], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{ RowBox[{"sec", "(", FractionBox["\[Pi]", "14"], ")"}], " ", RowBox[{"(", RowBox[{"x", "+", RowBox[{"sin", "(", FractionBox["\[Pi]", "14"], ")"}]}], ")"}]}], ")"}], " ", RowBox[{"cos", "(", FractionBox["\[Pi]", "14"], ")"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{3.396209732037717*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396209779723214*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 7", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962097918232527`*^9, 3.396209797339656*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Table"}]], "Input", CellChangeTimes->{{3.396209837426765*^9, 3.39620984532789*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates a list of \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]]\\) copies of \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\). \\n\ \\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"i\\\", \\\"TI\ \\\"], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates \ a list of the values of \\!\\(\\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\\) when \ \\!\\(\\*StyleBox[\\\"i\\\", \\\"TI\\\"]\\) runs from 1 to \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"max\\\", \ \\\"TI\\\"]]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) starts \ with \\!\\(\\*RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\"=\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]]}]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]], \\\",\\\", StyleBox[\\\"di\\\", \ \\\"TI\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) uses steps \ \\!\\(\\*StyleBox[\\\"di\\\", \\\"TI\\\"]\\). \\n\\!\\(\\*RowBox[{\\\"Table\\\ \", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \ \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) \ uses the successive values \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\ \"], StyleBox[\\\"1\\\", \\\"TR\\\"]]\\), \ \\!\\(\\*SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\ \"TR\\\"]]\\), \\!\\(\\*StyleBox[\\\"\[Ellipsis]\\\", \ \\\"TR\\\"]\\).\\n\\!\\(\\*RowBox[{\\\"Table\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"i\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"i\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\ \", RowBox[{StyleBox[\\\"j\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"j\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"j\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"]\\\"}]\\) gives a nested list. The list \ associated with \\!\\(\\*StyleBox[\\\"i\\\", \\\"TI\\\"]\\) is outermost. \ \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Table"]}]], "Print", "PrintUsage", CellChangeTimes->{3.396209860317333*^9}, CellTags->"Info3396184659-9476632"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"triangularnumbers", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], "k"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "10"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "3", ",", "6", ",", "10", ",", "15", ",", "21", ",", "28", ",", "36", ",", "45", ",", "55"}], "}"}]], "Output", CellChangeTimes->{3.396210023859614*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sumofcubes", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], SuperscriptBox["k", "3"]}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "10"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "9", ",", "36", ",", "100", ",", "225", ",", "441", ",", "784", ",", "1296", ",", "2025", ",", "3025"}], "}"}]], "Output", CellChangeTimes->{3.3962101286923656`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], SuperscriptBox["k", "3"]}], "===", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], "k"}], ")"}], "2"]}]], "Input"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396210190827898*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{653, 700}, WindowMargins->{{Automatic, 202}, {95, Automatic}}, ShowSelection->True, Magnification->1.25, FrontEndVersion->"7.0 for Mac OS X PowerPC (32-bit) (November 11, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "Info3396182190-1141342"->{ Cell[1335, 51, 563, 13, 83, "Print", CellTags->"Info3396182190-1141342"]}, "Info3396182505-3705663"->{ Cell[3970, 155, 432, 11, 64, "Print", CellTags->"Info3396182505-3705663"]}, "Info3442140979-7902732"->{ Cell[5756, 218, 565, 13, 83, "Print", CellTags->"Info3442140979-7902732"]}, "Info3442140979-6828370"->{ Cell[8019, 299, 430, 11, 63, "Print", CellTags->"Info3442140979-6828370"]}, "Info3442140981-9562016"->{ Cell[25524, 902, 3886, 60, 197, "Print", CellTags->"Info3442140981-9562016"]}, "Info3396184659-9476632"->{ Cell[48785, 1731, 3884, 60, 291, "Print", CellTags->"Info3396184659-9476632"]} } *) (*CellTagsIndex CellTagsIndex->{ {"Info3396182190-1141342", 54567, 1868}, {"Info3396182505-3705663", 54676, 1871}, {"Info3442140979-7902732", 54786, 1874}, {"Info3442140979-6828370", 54896, 1877}, {"Info3442140981-9562016", 55006, 1880}, {"Info3396184659-9476632", 55119, 1883} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 456, 14, 57, "Subtitle"], Cell[1026, 38, 165, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[1216, 47, 116, 2, 33, "Input"], Cell[1335, 51, 563, 13, 83, "Print", CellTags->"Info3396182190-1141342"] }, Open ]], Cell[CellGroupData[{ Cell[1935, 69, 152, 3, 33, "Input"], Cell[2090, 74, 87, 1, 33, "Output"] }, Open ]], Cell[2192, 78, 202, 3, 32, "Text"], Cell[CellGroupData[{ Cell[2419, 85, 242, 8, 58, "Input"], Cell[2664, 95, 87, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2788, 101, 233, 6, 67, "Input"], Cell[3024, 109, 95, 2, 36, "Output"] }, Open ]], Cell[3134, 114, 67, 4, 51, "Text"], Cell[CellGroupData[{ Cell[3226, 122, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[3412, 131, 323, 12, 66, "Input"], Cell[3738, 145, 76, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3851, 151, 116, 2, 33, "Input"], Cell[3970, 155, 432, 11, 64, "Print", CellTags->"Info3396182505-3705663"] }, Open ]], Cell[CellGroupData[{ Cell[4439, 171, 185, 3, 53, "Input"], Cell[4627, 176, 109, 1, 33, "Output"] }, Open ]], Cell[4751, 180, 84, 2, 33, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4884, 188, 95, 1, 95, "Title"], Cell[4982, 191, 114, 1, 57, "Subtitle"], Cell[CellGroupData[{ Cell[5121, 196, 125, 2, 33, "Input"], Cell[5249, 200, 178, 3, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5464, 208, 145, 2, 83, "Section"], Cell[CellGroupData[{ Cell[5634, 214, 119, 2, 33, "Input"], Cell[5756, 218, 565, 13, 83, "Print", CellTags->"Info3442140979-7902732"] }, Open ]], Cell[CellGroupData[{ Cell[6358, 236, 154, 3, 33, "Input"], Cell[6515, 241, 166, 3, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6718, 249, 231, 6, 83, "Input"], Cell[6952, 257, 144, 3, 36, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[7145, 266, 94, 1, 83, "Section"], Cell[CellGroupData[{ Cell[7264, 271, 444, 14, 61, "Input"], Cell[7711, 287, 152, 3, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7900, 295, 116, 2, 33, "Input"], Cell[8019, 299, 430, 11, 63, "Print", CellTags->"Info3442140979-6828370"] }, Open ]], Cell[CellGroupData[{ Cell[8486, 315, 150, 3, 33, "Input"], Cell[8639, 320, 220, 5, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8896, 330, 299, 7, 33, "Input"], Cell[9198, 339, 759, 17, 33, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[10006, 362, 94, 1, 83, "Section"], Cell[CellGroupData[{ Cell[10125, 367, 390, 13, 61, "Input"], Cell[10518, 382, 191, 4, 55, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10746, 391, 193, 4, 33, "Input"], Cell[10942, 397, 244, 5, 33, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[11235, 408, 96, 1, 83, "Section"], Cell[CellGroupData[{ Cell[11356, 413, 1180, 35, 33, "Input"], Cell[12539, 450, 1046, 25, 111, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13622, 480, 303, 6, 36, "Input"], Cell[13928, 488, 148, 3, 33, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[14125, 497, 96, 1, 83, "Section"], Cell[CellGroupData[{ Cell[14246, 502, 255, 6, 61, "Input"], Cell[14504, 510, 233, 5, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14774, 520, 303, 9, 68, "Input"], Cell[15080, 531, 324, 10, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15441, 546, 275, 8, 67, "Input"], Cell[15719, 556, 354, 11, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16110, 572, 410, 13, 70, "Input"], Cell[16523, 587, 178, 3, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16738, 595, 393, 12, 68, "Input"], Cell[17134, 609, 301, 10, 69, "Output"] }, Open ]], Cell[17450, 622, 166, 4, 81, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[17653, 631, 94, 1, 83, "Section"], Cell[CellGroupData[{ Cell[17772, 636, 378, 9, 99, "Input"], Cell[18153, 647, 648, 21, 72, "Output"], Cell[18804, 670, 723, 24, 73, "Output"], Cell[19530, 696, 337, 7, 66, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19904, 708, 521, 17, 103, "Input"], Cell[20428, 727, 2086, 71, 343, "Output"], Cell[22517, 800, 2720, 87, 237, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[25286, 893, 94, 1, 83, "Section"], Cell[CellGroupData[{ Cell[25405, 898, 116, 2, 33, "Input"], Cell[25524, 902, 3886, 60, 197, "Print", CellTags->"Info3442140981-9562016"] }, Open ]], Cell[CellGroupData[{ Cell[29447, 967, 327, 9, 66, "Input"], Cell[29777, 978, 313, 6, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30127, 989, 336, 10, 66, "Input"], Cell[30466, 1001, 295, 7, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30798, 1013, 134, 2, 38, "Input"], Cell[30935, 1017, 295, 7, 33, "Output"] }, Open ]], Cell[31245, 1027, 363, 11, 68, "Input"], Cell[CellGroupData[{ Cell[31633, 1042, 30, 0, 33, "Input"], Cell[31666, 1044, 146, 3, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[31849, 1052, 268, 7, 53, "Input"], Cell[32120, 1061, 637, 15, 33, "Output"] }, Open ]], Cell[32772, 1079, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[32876, 1088, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[33062, 1097, 323, 12, 66, "Input"], Cell[33388, 1111, 94, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[33519, 1118, 144, 3, 33, "Input"], Cell[33666, 1123, 146, 3, 33, "Output"] }, Open ]], Cell[33827, 1129, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[33931, 1138, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[34117, 1147, 155, 4, 40, "Input"], Cell[34275, 1153, 107, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34419, 1159, 121, 3, 40, "Input"], Cell[34543, 1164, 105, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34685, 1170, 121, 3, 40, "Input"], Cell[34809, 1175, 105, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34951, 1181, 134, 4, 40, "Input"], Cell[35088, 1187, 104, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35229, 1193, 353, 9, 40, "Input"], Cell[35585, 1204, 73, 1, 33, "Output"] }, Open ]], Cell[35673, 1208, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[35777, 1217, 163, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[35965, 1226, 140, 4, 66, "Input"], Cell[36108, 1232, 113, 3, 53, "Output"] }, Open ]], Cell[36236, 1238, 203, 6, 32, "Text"], Cell[CellGroupData[{ Cell[36464, 1248, 235, 8, 65, "Input"], Cell[36702, 1258, 119, 3, 58, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36858, 1266, 208, 7, 65, "Input"], Cell[37069, 1275, 277, 9, 53, "Output"] }, Open ]], Cell[37361, 1287, 330, 11, 51, "Text"], Cell[CellGroupData[{ Cell[37716, 1302, 342, 12, 68, "Input"], Cell[38061, 1316, 83, 2, 39, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38181, 1323, 325, 11, 68, "Input"], Cell[38509, 1336, 224, 8, 69, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38770, 1349, 108, 3, 33, "Input"], Cell[38881, 1354, 101, 2, 33, "Output"] }, Open ]], Cell[38997, 1359, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[39101, 1368, 160, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[39286, 1377, 180, 6, 58, "Input"], Cell[39469, 1385, 521, 19, 72, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40027, 1409, 134, 2, 33, "Input"], Cell[40164, 1413, 596, 22, 73, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40797, 1440, 180, 6, 57, "Input"], Cell[40980, 1448, 1820, 65, 334, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[42837, 1518, 180, 6, 57, "Input"], Cell[43020, 1526, 2390, 78, 269, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45447, 1609, 291, 9, 71, "Input"], Cell[45741, 1620, 2621, 86, 296, "Output"] }, Open ]], Cell[48377, 1709, 67, 4, 63, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[48481, 1718, 163, 5, 40, "Subsubtitle"], Cell[CellGroupData[{ Cell[48669, 1727, 113, 2, 51, "Input"], Cell[48785, 1731, 3884, 60, 291, "Print", CellTags->"Info3396184659-9476632"] }, Open ]], Cell[CellGroupData[{ Cell[52706, 1796, 271, 8, 102, "Input"], Cell[52980, 1806, 209, 5, 51, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[53226, 1816, 291, 9, 102, "Input"], Cell[53520, 1827, 222, 5, 51, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[53779, 1837, 294, 10, 107, "Input"], Cell[54076, 1849, 73, 1, 51, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)