(* 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[ 20694, 741] NotebookOptionsPosition[ 17649, 642] NotebookOutlinePosition[ 18032, 659] CellTagsIndexPosition[ 17989, 656] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 7 Sums, Series, Products and Limits", "Section", CellChangeTimes->{3.4610864122323008`*^9}], Cell["\<\ In this lesson you will learn about the built-in functions related to sums, \ series, products and limits.\ \>", "Text"], Cell[CellGroupData[{ Cell["Sums", "Subsection"], Cell[TextData[{ "We have already use ", StyleBox["Total", "Input"], " to sum up the entries in a list. The ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Sum", "Input"], " serves a similar purpose but has syntax a bit like ", StyleBox["Table", "Input"], ". Here are some examples:" }], "Text", CellChangeTimes->{{3.4610865119742813`*^9, 3.4610866354370747`*^9}, { 3.461086746826667*^9, 3.461086751610208*^9}}], Cell[TextData[{ "The sum of the first ", StyleBox["20", "Input"], " integers, ", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "20"], "i"}], TraditionalForm]]], ":" }], "Text", CellChangeTimes->{{3.461086650981614*^9, 3.461086651476705*^9}, 3.461086765373137*^9, {3.4611044508879023`*^9, 3.461104472951865*^9}}], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", "20"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461086654097816*^9, 3.46108667135394*^9}}], Cell["The sum of the squares of the first few odd integers :", "Text", CellChangeTimes->{{3.461086700622843*^9, 3.461086714268046*^9}, 3.4610867692363777`*^9}], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"i", "^", "2"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "20", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461086678151414*^9, 3.4610866967737513`*^9}}], Cell["The sum of some sine function values", "Text", CellChangeTimes->{{3.461086778717218*^9, 3.4610868190942717`*^9}}], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"Sin", "[", "i", "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "11"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.46108682151127*^9, 3.461086841320159*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can evaluate many sums in closed form. For example, it knows the formula \ for the sum of the first ", StyleBox["n", "Input"], " integers." }], "Text", CellChangeTimes->{3.4610865179794416`*^9}], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "n"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "Since ", StyleBox["Mathematica", FontSlant->"Italic"], " always gives exact results, you might get large fractions" }], "Text"], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "20"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4610871430455637`*^9, 3.4610871454989977`*^9}}], Cell[TextData[{ "Rather than using ", StyleBox["N[Sum[...]]", "Input", FontWeight->"Bold"], " or ", StyleBox["Sum[...]//N", "Input", FontWeight->"Bold"], ", you can use the function ", StyleBox["NSum", "Input", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Sum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "20"}], "}"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.461087147487802*^9, 3.46108714893116*^9}}], Cell[BoxData[ RowBox[{"NSum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "20"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4610871509569473`*^9, 3.461087153754731*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[" Practice", "Subsection", CellChangeTimes->{ 3.4610871692213717`*^9, {3.4610893010558863`*^9, 3.46108930301775*^9}}], Cell[TextData[{ "1) Calculate ", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "20"], FractionBox["1", RowBox[{"n", "!"}]]}], TraditionalForm]]], ", a quantity that should be near ", "\[ExponentialE]. How close are these two numbers?" }], "Text", CellChangeTimes->{{3.461089320770793*^9, 3.461089437555436*^9}, { 3.4610895185401077`*^9, 3.4610895188132153`*^9}}], Cell[TextData[{ "2) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find a formula for the sum of the cubes of the first ", StyleBox["n", "Input"], " natural numbers." }], "Text", CellChangeTimes->{{3.46108955103293*^9, 3.4610895844106083`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Series", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " also handles series, the \"infinite\" version of sums. Here is a \ calculation showing that ", StyleBox["Mathematica", FontSlant->"Italic"], " knows that\n1-", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "+", FractionBox["1", "3"], "-", FractionBox["1", "4"], "+"}], "..."}], "=", RowBox[{"ln", " ", RowBox[{"(", "2", ")"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.461089625400271*^9, 3.4610897068183603`*^9}}], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "^", RowBox[{"(", RowBox[{"n", "+", "1"}], ")"}]}], "/", "n"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "Infinity"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.46108972207393*^9, 3.461089726559566*^9}}], Cell["\<\ Here is what happens if you try to compute a series that does not converge:\ \>", "Text"], Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"1", "/", "i"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "Infinity"}], "}"}]}], "]"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellChangeTimes->{{3.461089793334025*^9, 3.461089799042265*^9}}], Cell["1) Calculate the following exactly:", "Text", CellChangeTimes->{{3.461089807147084*^9, 3.461089831066063*^9}}], Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], FractionBox["1", RowBox[{"n", "!"}]]}]], "Input", CellChangeTimes->{{3.461089843926291*^9, 3.461089862184022*^9}}], Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], FractionBox["1", SuperscriptBox["n", "2"]]}]], "Input", CellChangeTimes->{{3.461089870484188*^9, 3.461089894423587*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"2", ")"}], " ", "How", " ", "close", " ", "is", " ", "the", " ", "sum", " ", "of", " ", "the", " ", "first", " ", "hundred", " ", "terms", " ", "of", " ", "the", " ", "series", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], RowBox[{ FractionBox["1", SuperscriptBox["n", "2"]], "to", " ", "the", " ", "exact", " ", "value", " ", "of", " ", "the", " ", RowBox[{"series", "?"}]}]}]}]], "Text", CellChangeTimes->{{3.461105457975884*^9, 3.461105470607511*^9}}, FontFamily->"Times New Roman", FontColor->GrayLevel[0]] }, Open ]], Cell[CellGroupData[{ Cell["Products", "Subsection"], Cell[TextData[{ "Products can be calculated with the function ", StyleBox["Product,", "Input", FontWeight->"Bold"], " having the same syntax as ", StyleBox["Sum", "Input"], ". Here is a lengthy way to compute 10!" }], "Text", CellChangeTimes->{ 3.461089947028635*^9, {3.461104600350745*^9, 3.461104660486071*^9}, { 3.4611047343087683`*^9, 3.461104784983595*^9}, {3.46110482997165*^9, 3.461104854062634*^9}}], Cell[BoxData[ RowBox[{"Product", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4611045304704323`*^9, 3.4611045385785227`*^9}}], Cell["Here's the fast way:", "Text", CellChangeTimes->{{3.461104559351912*^9, 3.4611045713109207`*^9}}], Cell[BoxData[ RowBox[{"10", "!"}]], "Input"], Cell[TextData[{ "Here is the product of the first 10 prime numbers. (", StyleBox["Prime[i]", "Input"], " is the ", StyleBox[Cell[BoxData[ FormBox[ SuperscriptBox["i", "th"], TraditionalForm]], "Input"], "Input"], StyleBox[" ", "Input"], "prime number: ", StyleBox["Prime[1]=2", "Input"], "; ", StyleBox["Prime[2]=3", "Input"], "; ", StyleBox["Prime[3]=5", "Input"], ", etc.)" }], "Text", CellChangeTimes->{ 3.461089947028635*^9, {3.461104600350745*^9, 3.461104660486071*^9}, { 3.4611047343087683`*^9, 3.4611048137004633`*^9}, 3.461104865035474*^9}], Cell[BoxData[ RowBox[{"Product", "[", RowBox[{ RowBox[{"Prime", "[", "i", "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "10"}], "}"}]}], "]"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Limits", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can compute limits, both approaching a finite value and approaching ", StyleBox["\[PlusMinus]\[Infinity]", "Input"], ". By default the limit is approached from above (unless the limit involves \ \[Infinity]), but you can change the direction with an option. " }], "Text", CellChangeTimes->{{3.416962052237522*^9, 3.4169620674912043`*^9}, { 3.448148096734157*^9, 3.4481481085108347`*^9}, 3.461104984383595*^9}], Cell[BoxData[ RowBox[{"?", "Limit"}]], "Input"], Cell["Here are some examples of limits:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Limit", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "/", "x"}], ",", RowBox[{"x", "->", "0"}]}], "]"}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{3.448148049558178*^9}], Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"3", SuperscriptBox["x", "4"]}], "-", "2"}], SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "-", "2"}], ")"}], "2"]], ",", RowBox[{"x", "\[Rule]", "Infinity"}]}], "]"}]], "Input", CellChangeTimes->{{3.461105001210992*^9, 3.461105064139213*^9}}], Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"Exp", "[", RowBox[{"-", "x"}], "]"}], ",", " ", RowBox[{"x", " ", "->", " ", "Infinity"}]}], "]"}]], "Input", CellChangeTimes->{{3.461105226280261*^9, 3.461105227441148*^9}}], Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", RowBox[{"x", "\[Rule]", "Infinity"}]}], "]"}]], "Input", CellChangeTimes->{{3.461105169316383*^9, 3.461105179056571*^9}}], Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"1", "/", "x"}], ",", RowBox[{"x", "\[Rule]", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.461105182731098*^9, 3.461105192168894*^9}}], Cell[TextData[{ "Some limits do not exist because the values do not converge (rather than \ going off to \[PlusMinus]\[Infinity]). In those cases, ", StyleBox["Mathematica", FontSlant->"Italic"], " sometimes will give you an interval as answer, namely the interval which \ consists of the values taken as the variable comes close to the specified \ value." }], "Text", CellChangeTimes->{{3.416962136445306*^9, 3.416962151266927*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"x", " ", "\[Rule]", " ", "Infinity"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Interval", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}], "]"}]], "Output", CellChangeTimes->{3.416962157572707*^9}] }, Open ]], Cell["Pretty cool!", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection"], Cell[BoxData[ RowBox[{ RowBox[{"1", ")"}], " ", "Calculate", StyleBox[ RowBox[{" ", " "}]], RowBox[{ UnderscriptBox["lim", RowBox[{"n", "\[Rule]", "\[Infinity]"}]], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", FractionBox["a", "n"]}], ")"}], "n"], "."}]}]}]], "Text", CellChangeTimes->{{3.4611054168975353`*^9, 3.461105427236074*^9}}, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[TextData[{ "2) Calculate ", Cell[BoxData[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "4"}]], FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", " ", "x"}], "-", "8"}], RowBox[{"x", "-", "4"}]]}]], CellChangeTimes->{{3.461105612180943*^9, 3.461105614141696*^9}, { 3.461105700470907*^9, 3.461105758536686*^9}}], "." }], "Text", CellChangeTimes->{{3.4611055705906897`*^9, 3.461105597807445*^9}, { 3.461105772183176*^9, 3.461105780251246*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Homework", "Subsection", CellChangeTimes->{{3.461105827225432*^9, 3.461105831044677*^9}}], Cell[TextData[{ "1) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to determine the following limits. Be careful to check whether the limits \ exist.\n(a) ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "\[Infinity]"}]], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox["1", "x"]}], ")"}], "x"]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.449231336640122*^9, 3.4492313384541903`*^9}, { 3.4611059994769783`*^9, 3.461106000755149*^9}}, FontWeight->"Plain"], Cell[TextData[{ "(b) ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[Rule]", "0"}]], FractionBox[ RowBox[{ SqrtBox[ RowBox[{"4", "+", "x"}]], "-", "2"}], "x"]}], TraditionalForm]]] }], "Text", CellDingbat->None, CellChangeTimes->{{3.4492313441181593`*^9, 3.449231347677897*^9}}, FontWeight->"Plain"], Cell[TextData[{ "2) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to calculate the following:\n(a) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], FractionBox["1", SuperscriptBox["n", "3"]]}], FontSize->16], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.417733323845963*^9, 3.4177333306696243`*^9}, { 3.449231465088401*^9, 3.449231465822597*^9}, {3.461106018022595*^9, 3.461106019106988*^9}}, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[{ "(b) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "100"], FractionBox["1", SuperscriptBox["n", "3"]]}], FontSize->16], TraditionalForm]]] }], "Subsubsection", CellDingbat->None, CellChangeTimes->{{3.417733323845963*^9, 3.4177333306696243`*^9}, { 3.449231470575086*^9, 3.449231482462484*^9}}, FontWeight->"Plain"], Cell["\<\ (c) What is the difference (in decimal representation) between the numbers \ calculated in (a) and (b)?\ \>", "Text", CellDingbat->None, CellChangeTimes->{{3.417733323845963*^9, 3.4177333306696243`*^9}, { 3.449231486966992*^9, 3.449231495054565*^9}}, FontWeight->"Plain"], Cell[TextData[{ "3) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to calculate the following:\n(a) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox[ StyleBox["1", FontSize->12], "2"], "\[Times]", FractionBox["3", "4"], "\[Times]", FractionBox["5", "6"], "\[Times]", FractionBox["7", "8"], "\[Times]"}], FontSize->16], TraditionalForm]]], "...", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"\[Times]", FractionBox["97", "98"], "\[Times]", FractionBox["99", "100"]}], FontSize->16], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.4492317398502407`*^9, 3.449231742048436*^9}, { 3.4611060361453533`*^9, 3.461106037250887*^9}}, FontWeight->"Plain"], Cell[TextData[{ "(b) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ FractionBox[ StyleBox["1", FontSize->12], "2"], "\[Times]", FractionBox["3", "4"], "\[Times]", FractionBox["5", "6"], "\[Times]", FractionBox["7", "8"], "\[Times]"}], "..."}], FormBox[ StyleBox[ RowBox[{"\[Times]", FractionBox["997", "998"], "\[Times]", FractionBox["999", "1000"]}], FontSize->16], TraditionalForm]}], FontSize->16], TraditionalForm]]] }], "Text", CellDingbat->None, CellChangeTimes->{{3.44923174694471*^9, 3.4492317503842783`*^9}, 3.449231872514783*^9, 3.44923198706409*^9}, FontWeight->"Plain"], Cell["\<\ (c) What is the difference (in decimal representation) between the numbers \ calculated in (a) and (b)?\ \>", "Text", CellDingbat->None, CellChangeTimes->{{3.4492317541120768`*^9, 3.449231756624103*^9}}, FontWeight->"Plain"], Cell[TextData[{ "4) Can ", StyleBox["Sum", "Input"], " be used to calculate the sum of a sequence of vectors? " }], "Text", CellChangeTimes->{{3.461106195941146*^9, 3.4611062519476337`*^9}, { 3.46110631672115*^9, 3.4611063509631577`*^9}, 3.461106417875456*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{780, 681}, WindowMargins->{{117, Automatic}, {103, Automatic}}, ShowSelection->True, Magnification->1., FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 108, 1, 67, "Section"], Cell[701, 26, 130, 3, 26, "Text"], Cell[CellGroupData[{ Cell[856, 33, 26, 0, 34, "Subsection"], Cell[885, 35, 450, 13, 41, "Text"], Cell[1338, 50, 376, 12, 33, "Text"], Cell[1717, 64, 194, 5, 27, "Input"], Cell[1914, 71, 164, 2, 26, "Text"], Cell[2081, 75, 241, 6, 27, "Input"], Cell[2325, 83, 120, 1, 26, "Text"], Cell[2448, 86, 225, 6, 27, "Input"], Cell[2676, 94, 265, 8, 26, "Text"], Cell[2944, 104, 138, 4, 27, "Input"], Cell[3085, 110, 149, 5, 26, "Text"], Cell[3237, 117, 258, 7, 27, "Input"], Cell[3498, 126, 260, 11, 26, "Text"], Cell[3761, 139, 286, 8, 27, "Input"], Cell[4050, 149, 257, 7, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[4344, 161, 125, 2, 34, "Subsection"], Cell[4472, 165, 435, 13, 36, "Text"], Cell[4910, 180, 263, 8, 26, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[5210, 193, 28, 0, 34, "Subsection"], Cell[5241, 195, 574, 19, 51, "Text"], Cell[5818, 216, 360, 11, 27, "Input"], Cell[6181, 229, 99, 2, 26, "Text"], Cell[6283, 233, 169, 5, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[6489, 243, 96, 1, 34, "Subsection"], Cell[6588, 246, 117, 1, 26, "Text"], Cell[6708, 249, 222, 6, 54, "Input"], Cell[6933, 257, 228, 6, 53, "Input"], Cell[7164, 265, 627, 15, 53, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[7828, 285, 30, 0, 34, "Subsection"], Cell[7861, 287, 425, 11, 26, "Text"], Cell[8289, 300, 203, 5, 27, "Input"], Cell[8495, 307, 104, 1, 26, "Text"], Cell[8602, 310, 45, 1, 27, "Input"], Cell[8650, 313, 572, 18, 44, "Text"], Cell[9225, 333, 166, 5, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[9428, 343, 28, 0, 34, "Subsection"], Cell[9459, 345, 483, 9, 41, "Text"], Cell[9945, 356, 48, 1, 27, "Input"], Cell[9996, 359, 49, 0, 26, "Text"], Cell[10048, 361, 244, 7, 43, "Input"], Cell[10295, 370, 412, 13, 55, "Input"], Cell[10710, 385, 245, 6, 27, "Input"], Cell[10958, 393, 208, 5, 27, "Input"], Cell[11169, 400, 201, 5, 27, "Input"], Cell[11373, 407, 440, 9, 41, "Text"], Cell[CellGroupData[{ Cell[11838, 420, 159, 4, 27, "Input"], Cell[12000, 426, 171, 5, 27, "Output"] }, Open ]], Cell[12186, 434, 28, 0, 26, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[12251, 439, 30, 0, 34, "Subsection"], Cell[12284, 441, 452, 15, 42, "Text"], Cell[12739, 458, 516, 16, 39, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[13292, 479, 96, 1, 34, "Subsection"], Cell[13391, 482, 566, 18, 53, "Text"], Cell[13960, 502, 367, 14, 40, "Text"], Cell[14330, 518, 546, 18, 55, "Text"], Cell[CellGroupData[{ Cell[14901, 540, 427, 15, 39, "Subsubsection"], Cell[15331, 557, 286, 7, 26, "Text"], Cell[15620, 566, 748, 27, 55, "Text"], Cell[16371, 595, 728, 26, 40, "Text"], Cell[17102, 623, 237, 6, 26, "Text"], Cell[17342, 631, 267, 6, 26, "Text"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)