(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 13659, 372]*) (*NotebookOutlinePosition[ 14311, 395]*) (* CellTagsIndexPosition[ 14267, 391]*) (*WindowFrame->Normal*) Notebook[{ Cell["Sequences", "Subtitle"], Cell["Goals of this lab", "Section"], Cell["\<\ 1. To learn what sequences are --- how they are alike and \ different from other functions. 2. To learn what is meant by the graph of a sequence. 3. To learn what is meant by \"the limit of a sequence\" and be able to \ determine this limit when it exists. 4. To learn the difference between an explicit and a recursion formula for a \ sequence.\ \>", "Text"], Cell["What are sequences?", "Section"], Cell[TextData[{ "We are all accustomed to functions whose domains include all positive real \ numbers --- that is, the interval ", Cell[BoxData[ \(TraditionalForm\`0\ < \ x\ < \ \(\(\[Infinity]\)\(.\)\)\)]], " Any polynomial includes this interval in its domain. The sine, cosine, \ exponential and logarithmic functions include it as well. What if, however, \ we suddenly restricted ourselves to inputs which were positive integers? \ These continuous functions, like the sine function whose graph is produced by \ the next cell" }], "Text"], Cell[BoxData[ \(Plot[Sin[x], {x, 0, 20}]\)], "Input"], Cell[TextData[{ "would become a ", StyleBox["sequences", FontSlant->"Italic"], ", infinite lists of discrete numbers. For instance, the next cell \ produces the values of the sine function evaluated at the first twenty \ positive integers:" }], "Text"], Cell[BoxData[ \(p = Table[Sin[b], {b, 1, 20}]\)], "Input"], Cell[TextData[{ "The term ", StyleBox["sequence", FontSlant->"Italic"], " is a new one to us in this course. Whenever you consider a function with \ domain the positive integers, you are considering a sequence. That makes a \ sequence just an infinitely-long list of numbers, one number corresponding to \ each of the positive integers. The only thing that distinguishes a sequence \ from other kinds of functions is its domain. Whereas one generally writes\n\t\ ", Cell[BoxData[ \(TraditionalForm\`\(\(f(x)\ = \ x\^2\ - \ x\ + \ 7\)\(,\)\)\)]], "\nor some such formula for a function, it is common to use ", StyleBox["n", FontSlant->"Italic"], " as the independent variable (and ", StyleBox["a", FontSlant->"Italic"], " as the name of the function) when indicating a sequence, as in\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(a(n)\ = \ n\^2\ - \ n\ + \ 7\)\(,\)\)\)]], "\nor, perhaps even more commonly,\n\t", Cell[BoxData[ \(TraditionalForm\`a\_n\ = \ \(\(n\^2\)\(\ \)\(-\)\(\ \)\(n\)\(\ \)\(+\ \)\(\ \)\(7.\)\(\ \)\)\)]] }], "Text"], Cell[TextData[{ "Since sequences are functions on a particular domain, it is possible to \ plot graphs of sequences just as we plot graphs of other functions. The \ sequence\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n\ = \ \ sin\ n\)\(,\)\)\)]], "\nwhen viewed in the viewing window 1 \[LessSlantEqual] n \ \[LessSlantEqual] 20, has the following graph:" }], "Text"], Cell[BoxData[ \(ListPlot[Table[Sin[b], {b, 1, 20}], PlotStyle -> PointSize[0.02]]\)], "Input"], Cell[TextData[{ "The ", StyleBox["x", FontSlant->"Italic"], "-coordinates of these points are the positive integers 1, 2, ..., 20, \ while the ", StyleBox["y", FontSlant->"Italic"], "-coordinates are the values of the sine function at these inputs. Can you \ visualize the full sine function connecting up these points? The next cell \ plots both sin(x) and the sequence:" }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\n", \(DisplayTogether[\[IndentingNewLine]Plot[Sin[x], {x, 0, 20}, PlotRange \[Rule] All, PlotStyle \[Rule] Hue[ .4]], \n ListPlot[Table[Sin[n], {n, 1, 20}], PlotStyle \[Rule] Hue[ .8]], \n Prolog \[Rule] AbsolutePointSize[6]\[IndentingNewLine]]\)}], "Input"], Cell[CellGroupData[{ Cell["Exercise 1:", "Subsubsection"], Cell[TextData[{ "Write the first ten terms --- that is, ", Cell[BoxData[ \(TraditionalForm\`a\_1, \ a\_2, \ a\_3, \ ... , \ a\_10\)]], "--- of the sequence whose explicit formula is given by\n\t", Cell[BoxData[ \(TraditionalForm\`a\_n\ = \ \(\(\(2 n\)\/\(2\ + \ n\)\)\(.\)\)\)]], "\nThinking of the above expression as a function (a sequence), what is the \ domain of this function taken to be? Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to produce a graph of the sequence for 1 \[LessSlantEqual] n \ \[LessSlantEqual] 10." }], "Text"] }, Closed]], Cell["Limits of sequences", "Section"], Cell[TextData[{ "We have grown accustomed to taking limits when working with functions. \ The idea does not translate so easily to sequences: the expression\n \ lim ", StyleBox["f(x)", FontSlant->"Italic"], "\n ", StyleBox["x", FontSlant->"Italic"], " \[Rule] ", StyleBox["c", FontSlant->"Italic"], " \nonly makes sense if, in the domain of ", StyleBox["f", FontSlant->"Italic"], ", one can choose ", StyleBox["x", FontSlant->"Italic"], "-values (both from the left and the right) so they get arbitrarily close \ to ", StyleBox["x = c", FontSlant->"Italic"], ". Since the domain of a sequence is always the positive integers, an \ expression like\n lim ", Cell[BoxData[ \(TraditionalForm\`a\_n\)]], "\n ", StyleBox["n", FontSlant->"Italic"], " \[Rule] 32", StyleBox["\n", FontSlant->"Italic"], StyleBox["generally does not make sense", FontWeight->"Bold"], " --- one cannot choose her integers ", StyleBox["n", FontSlant->"Italic"], " arbitrarily close to 32, because the closet she can get (without actually \ making ", StyleBox["n", FontSlant->"Italic"], " = 32) is still a distance of one away!" }], "Text"], Cell[CellGroupData[{ Cell["Exercise 2:", "Subsubsection"], Cell["\<\ Starting from the definition of derivative, explain why it makes no \ sense to talk about the derivative of a sequence.\ \>", "Text"] }, Closed]], Cell[TextData[{ "The one type of limit that still makes sense for a sequence is\n", " lim ", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n\)\(.\)\)\)]], "\n ", StyleBox["n", FontSlant->"Italic"], " \[Rule] ", StyleBox["\[Infinity]", FontSlant->"Italic"], "\nAs with other limits we have seen, the \"limit of a sequence\" (as the \ above is called) may either exist (another common term for this is to say the \ sequence ", StyleBox["converges", FontSlant->"Italic"], ") or it may not (i.e., the sequence may ", StyleBox["diverge", FontSlant->"Italic"], "). Consider the sequence\n\t", Cell[BoxData[ \(TraditionalForm\`a\_n\ = \ 1/n, \ \ \ \ n\ = \ 1, \ 2, \ 3, \ ... \)]], "\nThe next cell plots the graphs of both this sequence and the continuous \ function ", StyleBox["f(x) = ", FontSlant->"Italic"], "1", StyleBox["/x", FontSlant->"Italic"], ". Is it not clear that both the function and the sequence converge to \ zero as the arguments (", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["n", FontSlant->"Italic"], " respectively) go to infinity?" }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\n", \(DisplayTogether[\[IndentingNewLine]Plot[1/x, {x, 1, 20}, PlotRange \[Rule] {0, 1}, PlotStyle \[Rule] Hue[ .4]], \n ListPlot[Table[1/n, {n, 1, 20}], PlotStyle \[Rule] Hue[ .8]], \n Prolog \[Rule] AbsolutePointSize[6]\[IndentingNewLine]]\)}], "Input"], Cell[TextData[{ "In fact, this suggests a method for finding the limit of a sequence. \ Originally, we said that a sequence could be generated by a function whose \ domain is (0, \[Infinity]) by simply evaluating that function at the positive \ integers --- i.e., by restricting the domain of that function. If we can \ reverse the process --- that is, given a sequence, extend the domain to (0, \ \[Infinity]) and produce a differentiable function in the process --- then we \ may find the limit of the sequence by finding the limit as ", StyleBox["x", FontSlant->"Italic"], " \[RightArrow] \[Infinity] of the extended function." }], "Text"], Cell[CellGroupData[{ Cell["Exercise 3:", "Subsubsection"], Cell[TextData[{ "Below are the first few terms of a sequence. For each do the following:\n \ (i) Come up with an explicit formula for the sequence (", Cell[BoxData[ \(TraditionalForm\`a\_n\ = \ ... \)]], ").\n (ii) Find the limit of the sequence, if the sequence converges.\n\n\ (a) 1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, ...\n(b) 1/2, 4/3, 9/4, 16/5, \ 25/6, 36/7, 49/8, 64/9, 81/10, ...\n(c) 3/2, 4/4, 5/6, 6/8, 7/10, 8/12, \ 9/14, 10/16, 11/18, 12/20, 13/22, ...\n(d) 1/4, -2/5, 3/6, -4/7, 5/8, -6/9, \ 7/10, -8/11, 9/12, -10/13, ..." }], "Text"] }, Closed]], Cell["Questions to ponder:", "Section"], Cell["\<\ 1. Many sequences arise from restricting the domain of a function \ defined on (0, \[Infinity]). Is the last sentence true if we change the word \ \"many\" to \"all\"? Can you think of any examples in which it is not true? 2. If so many sequences arise from functions defined on a larger domain, \ what is the point of studying them? After all, we have already put (over the \ course of our mathematical lives) much effort into understanding functions. \ Why introduce this (somewhat) new concept?\ \>", "Text"], Cell["Recursion formulas for a sequence", "Section"], Cell[TextData[{ "Here is another sequence\n\t1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...\nIt \ would be hard to carry out Exercise 3 with this sequence. Take a minute to \ think about an explicit formula for ", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n\)\(.\)\)\)]], " Even though you probably cannot guess such a formula, you probably do \ recognize a pattern to the sequence. Most sequences, in fact, do not have \ discernible patterns. (For instance, if you can determine an exact pattern \ to the sequence of daily high temperatures in your area, that could make you \ rich!)\n\nThe pattern to the sequence above (known as the Fibonacci sequence, \ after Leonardo of Pisa, son of Bonacci) is that each term in the sequence is \ the sum of the previous two terms. Such a statement, expressed in symbols as\ \n\t", Cell[BoxData[ \(TraditionalForm\`\(\(a\_n\ = \ a\_\(n - 1\)\ + \ a\_\(n - 2\)\)\(,\)\)\)]], "\nis said to be a recursion formula. It is not the same as an explicit \ formula --- you cannot simply say, \"I want to know the 100th term in the \ sequence\", plug in ", StyleBox["n", FontSlant->"Italic"], " = 100, and get your answer. You can know the 100th term so long as you \ know the 99th and 98th, but to know the 99th you need to know the 98th and \ 97th, etc. In fact, such a recursion formula does not uniquely identify a \ sequence. Notice that, though the sequence\n\t3, -1, 2, 1, 3, 4, 7, 11, 18, \ 29, 47, 76, 123, ...\nis different from the one above, it obeys the exact \ same recursion formula. It is analogous to identifying a person named \ \"Smith\". This name sets her apart from others, but one needs more \ information about Smith (like a first name) to identify her specifically." }], "Text"], Cell[CellGroupData[{ Cell["Exercise 4:", "Subsubsection"], Cell[TextData[{ "Write out three distinct sequences (at least the first 10 terms) each of \ which satisfy the recursion formula\n\t", Cell[BoxData[ \(TraditionalForm\`a\_n\ = \ 3\ - \ \(\(a\_\(n - 1\)\)\(.\)\)\)]], "\nWhat additional information (in addition to the recursion formula) would \ be necessary to identify each of the sequences uniquely?\n\nTry to write an \ explicit formula for each of these three sequences.\n\n(Harder, but try it:) \ What information besides the recursion formula for the Fibonacci sequence \ would identify it uniquely?" }], "Text"] }, Closed]] }, FrontEndVersion->"4.2 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, CellGrouping->Manual, WindowSize->{520, 600}, WindowMargins->{{Automatic, 156}, {Automatic, 20}} ] (******************************************************************* Cached data follows. 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