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Professional software writers take the \ data contained in the power series and feed them into advanced approximation \ schemes to come up with incredibly accurate approximations of a function \ defined by a given power series.\nThis is true for everyday functions like ", Cell[BoxData[ \(f[x] = Sin[x]\)]], ". For instance, when ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluates ", Cell[BoxData[ \(Sin[x]\)]], ", it does not make measurements on triangles.\n", StyleBox["Mathematica", FontSlant->"Italic"], " takes the expansion of ", Cell[BoxData[ \(Sin[x]\)]], " in powers of ", Cell[BoxData[ \(x\)]], ",\n ", Cell[BoxData[ \(x - x\^3\/\(3!\) + x\^5\/\(5!\) - x\^7\/\(7!\) + x\^9\/\(9!\) + \)]], " ", Cell[BoxData[ \(\(\( ... \ \(+\ \((\(-1\))\)\^k\)\)\ x\^\(2\ k + 1\)\/\(\((2\ k + 1)\)!\) + \)\ ... \)]], " ;\nthen it extracts the data it needs from the expansion and creates its \ own efficient, accurate approximation of ", Cell[BoxData[ \(Sin[x]\)]], ". The actual process is both fascinating and technical. \nIf this catches \ your interest, you're a good candidate for going on to advanced work in \ numerical mathematics. " }], "SmallText"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["B.4) Convergence intervals for general power series", "Subsection", CellTags->"3.06.B4"], Cell["B.4.a.i)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ Given a specific power series, how do you try to estimate where the \ barriers are?\ \>", "Text"], Cell["Answer:", "Special1"], Cell["You can try eyeballing some plots.", "SmallText"] }, Closed]], Cell["B.4.a.ii) The Power Series Convergence Principle", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ How do you try to come up with more definitive information than you \ get from eyeball estimates?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "You can try to use the Power Series Convergence Principle.\nThis principle \ says that if ", Cell[BoxData[ \(f[x]\)]], " is defined by a power series\n ", Cell[BoxData[ \(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ a[3]\ x\^3\ + \)]], " ", Cell[BoxData[ \(\(\( ... \ \(+\ a[k]\)\)\ x\^k\ + \)\ ... \)]], " ,\nwhere the ", Cell[BoxData[ \(a[k]\)]], "'s are given constants, then you are guaranteed that the power series \ converges to ", Cell[BoxData[ \(f[x]\)]], " for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ", provided that the infinite list of individual terms\n ", Cell[BoxData[ \({a[0]\ , a[1]\ R, \ a[2]\ R\^2, a[3]\ R\^3, \ ... , \ a[k]\ R\^k, \ ... }\)]], " stays bounded (i.e., does not blow up to ", Cell[BoxData[ \(\(\[Infinity]\ \)\)]], "or down to ", Cell[BoxData[ \(\(\(-\[Infinity]\))\)\)]], "." }], "SmallText"] }, Closed]], Cell["B.4.b.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Go back to the function ", Cell[BoxData[ \(f[x]\)]], " whose expansion in powers of ", Cell[BoxData[ \(x\)]], " is\n ", Cell[BoxData[ \(1 - x + x\^2\/2\^2 - x\^3\/3\^2 + x\^4\/4\^2 + \)]], " ", Cell[BoxData[ \(\(\( ... \ \(+\((\(-1\))\)\^k\)\)\ x\^k\/k\^2\ + \)\ ... \)]], "\nUse the Power Series Convergence Principle to confirm that the power \ series converges to ", Cell[BoxData[ \(f[x]\)]], " for ", Cell[BoxData[ \(\(-\ 1\)\ < \ x\ < \ 1\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Take ", Cell[BoxData[ \(R\ = \ 1\)]], ", and look at the infinite list of individual terms\n ", Cell[BoxData[ \({1\ , \(-R\), R\^2\/2\^2, \(-\(R\^3\/3\^2\)\), R\^4\/4\^2, \ ... \ , \(\((\(-1\))\)\^k\) R\^k\/k\^2, \ ... }\)]], "\n ", Cell[BoxData[ \(\( = \ {1\ , \(-1\), \ 1\/2\^2, \(-\(1\/3\^2\)\), 1\/4\^2, \(-\(1\/5\^2\)\), \ \( ... \ \((\(-1\))\)\^k\/k\^2\), ... }\)\)]], ".\nAll the terms in this list are captured between ", Cell[BoxData[ \(\(-1\)\)]], " and ", Cell[BoxData[ \(1\)]], ". 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Cell[BoxData[ \(f[x]\)]], " for ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ". " }], "SmallText"] }, Closed]], Cell["B.4.b.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ Do you always have to use the Power Series Convergence Principle to \ determine a convergence interval for a power series?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "No.\nIf the power series comes from a differential equation as in B.3) \ above, you can inspect the differential equation to determine a convergence \ interval. \nIf you recognize a power series as an expansion of a function ", Cell[BoxData[ \(f[x]\)]], " you know about, then you can find convergence intervals by looking at \ complex singularities of ", Cell[BoxData[ \(f[x]\)]], ".\nCase in point:\nIf the power series\n ", Cell[BoxData[ \(1\ - \ x\^2\ + \ x\^4\ - \ x\^6\ + \ x\^8\ - \ x\^10\ + \)]], " ", Cell[BoxData[ \(\(\( ... \ \(+\ \((\(-1\))\)\^k\)\)\ x\^\(2 k\)\ + \)\ ... \)]], "\ncomes your way, you should recognize it as the expansion of \n ", Cell[BoxData[ \(f[x]\ = \ 1\/\(1\ + \ x\^2\)\)]], " \nin powers of ", Cell[BoxData[ \(x\)]], ". Since this function has complex singularities at ", Cell[BoxData[ \(i\)]], " and ", Cell[BoxData[ \(\(-i\)\)]], ", you know that ", Cell[BoxData[ \(R\ = \ 1\)]], ". Consequently the power series converges to \n ", Cell[BoxData[ \(f[x] = 1\/\(1\ + \ x\^2\)\)]], " \nfor ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ". You should reserve the Power Series Convergence Principle for \ situations you can't handle by another method." }], "SmallText"] }, Closed]], Cell["B.4.c)", "Subsubsection"], Cell[CellGroupData[{ Cell["What ideas underlie the Power Series Convergence Principle?", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "This is going to be quite technical. \nMaybe you can afford not to \ understand it fully. After you go on in mathematics, this little discussion \ will not seem so obtuse.\nThe Power Series Convergence Principle says that if \ ", Cell[BoxData[ \(f[x]\)]], " is defined by a power series\n ", Cell[BoxData[ \(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ a[3]\ x\^3\ + \)]], " ", Cell[BoxData[ \(\(\( ... \ \(+\ a[k]\)\)\ x\^k\ + \)\ ... \)]], "\nwhere the ", Cell[BoxData[ \(a[k]\)]], "'s are given constants, then the power series converges to ", Cell[BoxData[ \(f[x]\)]], " for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ", provided that the infinite list of individual terms\n ", Cell[BoxData[ \({a[0]\ , \ a[1]\ R, \ a[2]\ R\^2, \ a[3]\ R\^3, \ ... \ , \ a[k]\ R\^k, \ ... \ }\)]], " \nstays bounded. Saying that this list of individual terms stays bounded \ is the same as saying that you can get your hands on a number ", Cell[BoxData[ \(M\)]], " such that ", Cell[BoxData[ \(M\)]], " is larger than everything in the list of terms\n ", Cell[BoxData[ \({\(\( | a[0]\) | \)\ , \ \(\( | a[1]\ R\) | \), \ \(\( | a[2]\ R\^2\) | \), \ \(\( | a[3]\ R\^3\) | \), \ \(\( | a[4]\ R\^4\) | \), \)]], "\n ", Cell[BoxData[ \(\(\( ... \ , \ \(\( | a[k]\ R\^k\) | \), \ ... \)\ }\)\)]], ".\nNow take any ", Cell[BoxData[ \(x\)]], " with ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ", and note that\n ", Cell[BoxData[ \(\( | a[k]\ x\^k\) | \ \( \[LessEqual] \ \ | a[k]\ r\^k\) | \)]], ", \nwhere ", Cell[BoxData[ \(r\)]], " is any number with ", Cell[BoxData[ \(\( | x\) | \ \( < \ r\)\ < \ R\)]], ". ", Cell[BoxData[ \(\((r = \(R\ + \ x\)\/2\)\)]], " will do just fine.) Take another look:\n ", Cell[BoxData[ \(\( | a[k]\ x\^k\) | \ \[Ellipsis]\ \ | a[k]\ r\^k | \)]], " \n ", Cell[BoxData[ \(\( = \ \ \( | a[k]\ R\^k\) | \ \(r\^k/\) | R\^k | \)\)]], " \n ", Cell[BoxData[ \(\( = \ \ \( | a[k]\ R\^k\) | \ \((\(r/\) | \ R | )\)\^k\)\)]], " \n = ", Cell[BoxData[ \(\( | a[k]\ R\^k\) | \ t\^k\)]], " \n ", Cell[BoxData[ \(\( \[LessEqual] \ \ \ M\ t\^k\)\)]], " \nwhere ", Cell[BoxData[ \(t\ = \ \(r/\) | R | \)]], ".\nMake careful note of the fact that ", Cell[BoxData[ \(0\ < \ t\ < \ 1\)]], ".\nThis ensures that the series\n ", Cell[BoxData[ \(\(1\ + \ \ t\ + \ \ t\^2\ + \ \ t\^3\ + \ \ t\^4\ + \ t\^5\ + ... \)\ + \ t\^k\ + \ ... \)]], "\nis convergent to ", Cell[BoxData[ \(1\/\(1\ - \ t\)\)]], ". \nConsequently, the series\n ", Cell[BoxData[ \(\(M\ + \ M\ t\ + \ \ M\ t\^2\ + \ \ M\ t\^3\ + \ ... \)\ + \ M\ t\^k + \ ... \)]], "\nis convergent to ", Cell[BoxData[ \(M\/\(1\ - \ t\)\)]], ". \nHere is the situation:\nThe term ", Cell[BoxData[ \(\(\( | a[k] x\^k\) | \)\)]], " is under the corresponding term ", Cell[BoxData[ \(M\ t\^k\)]], "\nof the convergent series. This tells you that the power series\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ a[3]\ x\^3\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], "\nhas no choice but to converge at least as fast as the convergent series\n\ ", Cell[BoxData[ \(\(M\ + \ M\ t\ + \ M\ t\^2\ + \ M\ t\^3\ + \ M\ t\^4\ + \ ... \)\ + \ M\ t\^k\ + \ ... \)]], "\nThis tells you why\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ a[3]\ x\^3\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], "\nis convergent provided ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ". \nThe salient point of this detailed discussion is that when you \ go with ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ", then the power series converges faster than a multiple of a very \ well-known convergent series. " }], "SmallText"] }, Closed]] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ T.3) Using the Power Series Convergence Principle: The Ratio Test\ \>", "Subsection", CellTags->"3.06.T3"], Cell[TextData[{ "Given a power series\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], " ,\nyou can attempt to find some of its convergence intervals by using the \ Power Series Convergence Principle or by using \nThe Ratio Test. The Ratio \ Test says: \nIf, for some positive number ", Cell[BoxData[ \(R\)]], ", you can ascertain that \n ", Cell[BoxData[ \(\(\(|\)\(a[k\ + \ 1]\ R\^\(k\ + \ 1\)\)\)\/\(a[k]\ R\^k\) | \ \(\(\ \[LessEqual]\)\(\ \)\(1\)\)\)]], "\nfor all large ", Cell[BoxData[ SuperscriptBox["n", "\[Prime]", MultilineFunction->None]]], "s, then\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], "\nconverges for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], "." }], "Text"], Cell["T.3.a)", "Subsubsection"], Cell["\<\ Find convergence intervals of the following power series. Use the Power Series Convergence Principle directly, use the ratio test, or \ use any other method you like.\ \>", "Text"], Cell["T.3.a.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\(1\ - \ x\ + \ x\^2\ - \ x\^3\ + \ x\^4\ - \ x\^5\ + \ ... \)\ + \ \((\(-1\))\)\^n\ x\^n + \ ... \)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "\[Rule] By the Power Series Convergence Principle:\nThe power series is\n \ ", Cell[BoxData[ \(1 - x\)]], " + ", Cell[BoxData[ \(x\^2\)]], " - ", Cell[BoxData[ \(x\^3\)]], " + ", Cell[BoxData[ \(x\^4\)]], " - ", Cell[BoxData[ \(x\^5\)]], " + ... + ", Cell[BoxData[ \(\(\((\(-1\))\)\^n\) x\^n\)]], "+ ... \nFor ", Cell[BoxData[ \(R\ = \ 1\)]], ", the list of terms\n ", Cell[BoxData[ \({1, \ \(-R\)\ , \ R, \ \(-R\^3\), \ R\^4, \ ... \ , \ \((\(-1\))\)\^n\ R\^n, \ ... }\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\ \)\({\ 1, \ \(-1\), \ 1, \ \(-1\), \ 1, \ \(-1\), \ \(\(...\)\(\ \)\(\((\(,\)\((\(-1\))\))\)\^n\)\), \ \ ... }\)\)\)]], "\nstays bounded. \nThis guarantees that the power series converges for \n \ ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ". \n\n\[Rule] By the Ratio Test:\nThe power series is\n ", Cell[BoxData[ \(\(\(\(1\ - \ x\ + \ x\^2\ - \ x\^3\ + \ x\^4\ - \ x\^5\ + \ ... \)\ + \ \(\((\(-1\))\)\^n\) x\^n + \ ... \)\(\ \)\)\)]], " \nNote that ", Cell[BoxData[ \(\(\(\(|\)\(a[n]\)\)\(|\)\)\ = \ 1\)]], " for all ", Cell[BoxData[ \(n\)]], "'s. \nFor a positive number ", Cell[BoxData[ \(R\)]], ", look at \n ", Cell[BoxData[ \(\(\(\(|\)\(\(a[n\ + \ 1]\ R\^\(n\ + \ 1\)\)\/\(a[ n]\ R\^n\)\)\)\(|\)\)\ = \(R\^\(n\ + \ 1\)\/R\^n = \ R\)\)]], ".\nThese ratios are all ", Cell[BoxData[ \(\(\(\[LessEqual]\)\(\ \)\(1\)\)\)]], " for ", Cell[BoxData[ \(R\ = \ 1\)]], ".\nThis guarantees that the power series converges for \n ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ". \n\n", Cell[BoxData[ \( \[Rule] \)]], " The easiest way:\nThe power series is\n ", Cell[BoxData[ \(\(1\ - \ x\ + \ x\^2\ - \ x\^3\ + \ x\^4\ - \ x\^5\ + \ ... \)\ + \ \(\((\(-1\))\)\^n\) x\^n + \ ... \)]], " \nYou already know that this is the expansion of ", Cell[BoxData[ \(1\/\(1\ + \ x\)\)]], " in powers of ", Cell[BoxData[ \(x\)]], ", and that it converges for ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ".\nEach of these three answers is complete. \n\nExplaining the truth \ requires only one good argument. Explaining lies usually requires many long, \ involved points of view." }], "SmallText"] }, Closed]], Cell["T.3.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\(x\^2\/\(2\ 3\^2\) + x\^3\/\(3\ 3\^3\)\ + \ ... \)\ + \ x\^k\/\(k\ 3\^k\) + \ ... \)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Go with the ratio test: \nFor ", Cell[BoxData[ \(k \[GreaterEqual] 2\)]], ", read off:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[R, k, a];\)\), "\n", \(a[k_] = 1\/\(k\ 3\^k\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3\^\(-k\)\/k\)], "Output"] }, Open ]], Cell[TextData[{ "For a positive number ", Cell[BoxData[ \(R\)]], ", look at: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(ratio = Simplify[\(a[k + 1]\ R\^\(k + 1\)\)\/\(a[k]\ R\^k\)]\)], "Input",\ AspectRatioFixed->True], Cell[BoxData[ \(\(k\ R\)\/\(3 + 3\ k\)\)], "Output"] }, Open ]], Cell[TextData[{ "Look at the ratios for ", Cell[BoxData[ \(R = 3\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(ratio /. R \[Rule] 3\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(3\ k\)\/\(3 + 3\ k\)\)], "Output"] }, Open ]], Cell[TextData[{ "Evidently, these ratios are ", Cell[BoxData[ \(\(\(\[LessEqual]\)\(\ \)\(1\)\)\)]], " for ", Cell[BoxData[ \(R\ = \ 3\)]], ".\nThis guarantees that the power series converges for ", Cell[BoxData[ \(\(-3\)\ < \ x\ < \ 3\)]], "." }], "SmallText"], Cell[TextData[{ "Just for kicks, here's a second argument using the \nPower Series \ Convergence Principle:\nThe power series is:\n ", Cell[BoxData[ \(\(x\^2\/\(2\ 3\^2\) + x\^3\/\(3\ 3\^3\)\ + \ ... \)\ + \ x\^k\/\(k\ 3\^k\) + \ ... \)]], " \nFor ", Cell[BoxData[ \(R\ = \ 3\)]], ", the list of terms\n ", Cell[BoxData[ \({\(R\^2\/\(2\ 3\^2\)\)\(,\)\(R\^3/\((3\ 3\^3)\)\)\(,\)\(R\^4/\((4\ \ 3\^4)\)\)\(,\)\(\ \)\)]], " ", Cell[BoxData[ \(\(\(\(...\)\(,\)\(\ \)\(R\^k\/\(n\ 3\^k\)\)\(,\)\(\ \)\(...\)\)\(\ \)\ \(}\)\)\)]], "\n \n = ", Cell[BoxData[ \({\(3\^2\/\(2\ 3\^2\)\)\(,\)\(3\^3\/\(3\ 3\^3\)\)\(,\)\(\ \ \)\(3\^4\/\(4\ 3\^4\)\)\(,\)\(\ \)\)]], " ", Cell[BoxData[ \(\(\(\(...\)\(\ \)\(,\)\(3\^k\/\(n\ 3\^k\)\)\(,\)\(\ \)\(...\)\)\(\ \)\ \(}\)\)\)]], "\n \n ", Cell[BoxData[ \(\(\(=\)\(\ \)\({1\/2, \ 1\/3, 1\/4, \ 1\/5, \ ... \ , 1\/k, \ ... \ }\)\)\)]], "\nstays bounded.\nThis guarantees that the power series converges for ", Cell[BoxData[ \(\(-3\)\ < \ x\ < \ 3\)]], ". " }], "SmallText"] }, Closed]], Cell["T.3.a.iii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\(1\ + \ x\ + \ x\^2/2\^2\ + \ x\^3/3\^3\ + \ x\^4/4\^4\ + \ ... \)\ + \ x\^k/k\^k\ + \ ... \)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The Power Series Convergence Principle works like a charm. For any \ positive number ", Cell[BoxData[ \(x\ = \ R\)]], ", look at the list of terms\n {", Cell[BoxData[ \(\(1 + R + R\^2\/2\^2 + R\^3\/3\^3 + R\^4\/4\^4\ + \ ... \)\ + R\^k\/k\^k\ + \ ... \)]], "}.\nNotice that after the point at which k becomes larger than ", Cell[BoxData[ \(R\)]], ", the terms get smaller and smaller. \nTake a look at the case ", Cell[BoxData[ \(R = 9\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[k];\)\), "\n", \(\(ListPlot[Table[9\^n\/n\^n, {n, 1, 25}], PlotStyle \[Rule] {PointSize[0.02], Red}, AspectRatio \[Rule] 1\/GoldenRatio];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0380952 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MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "No matter what positive R you take, the list of terms\n {", Cell[BoxData[ \(\(1 + R + R\^2\/2\^2 + R\^3\/3\^3 + R\^4\/4\^4\ + \ ... \)\ + R\^k\/k\^k\ + \ ... \)]], "}\ncannot blow up to +", Cell[BoxData[ \(\[Infinity]\)]], " or down to ", Cell[BoxData[ \(\(-\[Infinity]\)\)]], ". \nThe Power Series Convergence Principle steps in to guarantee that no \ matter what positive ", Cell[BoxData[ \(R\)]], " you go with, this power series converges for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ".\nThe upshot:\nThis power series converges for -", Cell[BoxData[ \(\[Infinity]\)]], " < x < ", Cell[BoxData[ \(\[Infinity]\)]], "." }], "SmallText"] }, Closed]], Cell["T.3.a.iv)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\(\(1\)\(\ \)\(-\)\(\ \)\(2\^2\ x\)\(\ \)\(+\)\(\ \)\(3\^2\ x\^2\)\(\ \ \)\(-\)\(\ \)\(4\^2\ x\^3\)\(\ \)\(+\)\(\ \)\)\)]], " ", Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^k\)\)\ \((k\ + \ \ 1)\)\^2\ \ \ x\^k\)\(\ \)\(+\)\) ... \)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "By the Power Series Convergence Principle:\nTake any positive ", Cell[BoxData[ \(R\)]], " and look at the list of terms:\n {", Cell[BoxData[ \(\(\(1\)\(\ \)\(-\)\(\ \)\(\(2\^2\) R\)\(\ \)\(+\)\(\ \)\(3\^2\ R\^2\)\(\ \)\(-\)\(\ \)\(4\^2\ R\^3\)\(\ \ \)\(+\)\(\ \)\)\)]], " ", Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^k\)\)\ \((k\ + \ 1)\)\^2\ R\^k\ \)\(\ \)\(+\)\) ... \)]], "}.\nPut ", Cell[BoxData[ \(R\ = \ e\^t\)]], " to get\n {", Cell[BoxData[ \(\(\(1\)\(\ \)\(-\)\(\ \)\(\(2\^2\) e\^t\)\(\ \)\(+\)\(\ \)\(3\^2\ \(e\^2\)\^t\)\(\ \)\(-\)\(\ \)\(4\^2\ \ \(e\^3\)\^t\)\(\ \)\(+\)\(\ \)\)\)]], " ", Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^k\)\)\ \((k\ + \ 1)\)\^2\ \ e\^kt\)\(\ \)\(+\)\) ... \)]], "}.\nIf ", Cell[BoxData[ \(t\ < \ 0\)]], ", then \n ", Cell[BoxData[ \(lim\_\(t \[Rule] \[Infinity]\)\ \((\(-1\))\)\^k\ \((k\ + \ 1)\)\^2\ \ \ e\^kt\ = \ 0\)]], "\nbecause exponential growth dominates power growth.\nSo, if ", Cell[BoxData[ \(t\ < \ 0\)]], ", the list\n {", Cell[BoxData[ \(1\ - \ \(2\^2\) e\^t\ + \ 3\^2\ \(e\^2\)\^t\ - \ 4\^2\ \(e\^3\)\^t\ + \)]], Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^k\)\)\ \((k\ + \ 1)\)\^2\ \ e\^kt\)\(\ \)\(+\)\) ... \)]], "}\ncannot blow up to ", Cell[BoxData[ \(\(\(+\)\(\[Infinity]\)\(\ \)\)\)]], " or down to ", Cell[BoxData[ \(\(-\(\(\[Infinity]\)\(.\)\)\)\)]], "\nThe result:\nThe power series converges for ", Cell[BoxData[ \(\(-e\^t\)\ < \ x\ < \ e\^t\)]], " as long as ", Cell[BoxData[ \(t\ < \ 0\)]], ".\nBut as ", Cell[BoxData[ \(t\)]], " closes in on ", Cell[BoxData[ \(0\)]], " through negative numbers, ", Cell[BoxData[ \(e\^t\)]], " closes in on ", Cell[BoxData[ \(1\)]], ". \nThe upshot:\nThe power series converges for ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], ".\n\nHere's how to see this by use of the Ratio Test; " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[a, k, R];\)\), "\n", \(a[k_] = \((\(-1\))\)\^k\ \((k + 1)\)\^2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((\(-1\))\)\^k\ \((1 + k)\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(firstlook = Cancel[\(a[k + 1]\ R\^\(k + 1\)\)\/\(a[k]\ R\^k\)]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(\(\((2 + k)\)\^2\ R\)\/\((1 + k)\)\^2\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(secondlook = Together[ExpandAll[firstlook]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\(-4\)\ R - 4\ k\ R - k\^2\ R\)\/\(1 + 2\ k + k\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "So \n ", Cell[BoxData[ \(lim\_\(t \[Rule] \[Infinity]\) | \(a[k\ + \ 1]\ R\^\(k\ + \ 1\)\)\/\ \(a[k]\ R\^k\) | \ = \ R\)]], ". \nAs a result, if ", Cell[BoxData[ \(R\)]], " is any number with ", Cell[BoxData[ \(\(0\ \[LessEqual] \ R\)\ < \ 1\)]], ", then\n ", Cell[BoxData[ \(\(\(|\)\(\(a[k\ + \ 1]\ R\^\(k\ + \ 1\)\)\/\(a[ n]\ R\^n\)\)\) | \ \(\(<\)\(\ \)\(1\)\)\)]], " \nfor all large ", Cell[BoxData[ \(n\)]], "'s. The ratio test tells you that if ", Cell[BoxData[ \(R\)]], " is any number with ", Cell[BoxData[ \(\(0\ \[LessEqual] \ R\)\ < \ 1\)]], ", then this power series converges for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ". In short, this power series converges for \n ", Cell[BoxData[ \(\(-1\)\ < \ x\ < \ 1\)]], "." }], "SmallText"] }, Closed]], Cell["T.3.a.v)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\(1\ + \ 2\ x\ + \ \(2\^2\ x\^2\)\/\(2!\)\ + \ \(2\^3\ x\^3\)\/\(3!\)\ \ + \ ... \)\ + \ \(2\^k\ x\^k\)\/\(k!\)\ + \ ... \)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "This is a soft pitch.\nRecall that\n ", Cell[BoxData[ \(\(1\ + \ x\ + x\^2\/\(2!\) + x\^3\/\(3!\) + x\^4\/\(4!\)\ + \ ... \)\ + \ x\^k\/\(k!\)\ + \ ... \)]], "\nis the expansion of ", Cell[BoxData[ \(e\^x\)]], " in powers of ", Cell[BoxData[ \(x\)]], ". It converges to ", Cell[BoxData[ \(e\^x\)]], " for ", Cell[BoxData[ \(\(\(-\[Infinity]\)\(\ \)\(<\)\(\ \)\(x\)\(\ \)\(<\)\(\ \ \)\(\[Infinity]\)\(\ \)\)\)]], ".\nReplacing ", Cell[BoxData[ \(x\)]], " by ", Cell[BoxData[ \(2\ x\)]], " above shows that the power series in question is the expansion of ", Cell[BoxData[ \(e\^\(2 x\)\)]], " in powers of ", Cell[BoxData[ \(x\)]], ". It converges to ", Cell[BoxData[ \(e\^\(2 x\)\)]], " for -\[Infinity] < x < \[Infinity].\nConsequently, the power series\n \ ", Cell[BoxData[ \(\(1\ + \ 2\ x\ + \ \(2\^2\ x\^2\)\/\(2!\) + \(2\^3\ x\^3\)\/\(3!\) + \ \ ... \)\ + \(2\^k\ x\^k\)\/\(k!\)\ + \ ... \)]], ".\nconverges for ", Cell[BoxData[ \(\(-\[Infinity]\)\ < \ x\ < \ \(\(\[Infinity]\)\(.\)\)\)]] }], "SmallText"] }, Closed]], Cell["T.3.b) Why the ratio test works", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ How do you use the Power Series Convergence Principle to see why \ the ratio test works?\ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The Ratio Test says: \nIf for some positive number ", Cell[BoxData[ \(R\)]], ", you can ascertain that \n ", Cell[BoxData[ \(\(\(|\)\(a[k\ + \ v1]\ R\^\(k\ + \ 1\)\)\)\/\(a[k]\ R\^k\) | \ \(\(\ \[LessEqual]\)\(\ \)\(1\)\)\)]], "\nfor all large ", Cell[BoxData[ \(k\)]], "'s, then\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], "\nconverges for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ".\nSaying that \n ", Cell[BoxData[ \(\(\(|\)\(a[k\ + \ 1]\ R\^\(k\ + \ 1\)\)\)\/\(a[k]\ R\^k\) | \ \(\(\ \[LessEqual]\)\(\ \)\(1\)\)\)]], " \nfor all large ", Cell[BoxData[ \(k\)]], "'s is the same as saying that\n ", Cell[BoxData[ \(\(\(|\)\(a[ k\ + \ 1]\ R\^\(k + 1\)\)\(|\)\(\ \)\(\(\[LessEqual]\)\(\ \ \)\(\(|\)\(a[k]\ R\^k\)\)\)\(|\)\)\)]], "\nfor all large ", Cell[BoxData[ \(k\)]], "'s.\nThis tells you that the infinite list of individual terms\n ", Cell[BoxData[ \({a[1] R, \ a[2] R\^2, \ a[3] R\^3, \ ... \ , \ a[k] R\^k, \ ... \ }\)]], "\ncannot blow up to ", Cell[BoxData[ \(\[Infinity]\)]], " or down to ", Cell[BoxData[ \(\(-\[Infinity]\)\)]], ", because after some point in the list, the absolute values of the terms \ cannot increase.\nThe Basic Power Series Convergence Principle steps in to \ guarantee that\n ", Cell[BoxData[ \(\(a[0]\ + \ a[1]\ x\ + \ a[2]\ x\^2\ + \ ... \)\ + \ a[k]\ x\^k\ + \ ... \)]], "\nconverges for ", Cell[BoxData[ \(\(-R\)\ < \ x\ < \ R\)]], ".\nThat's all there is to it." }], "SmallText"] }, Closed]] }, Closed]], Cell["Give It a Try Problems", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "G.9) ", Cell[BoxData[ \(e\^x\)]], ", ", Cell[BoxData[ \(Sin[x]\)]], ", and ", Cell[BoxData[ \(Cos[x]\)]], " from the advanced viewpoint*" }], "Subsection", CellTags->"3.06.G9"], Cell[TextData[{ "By now most of you are becoming tired of reading that\n ", Cell[BoxData[ \(e\^x\ = \ \(1\ + \ x\ + x\^2\/2 + x\^3\/\(3!\)\ + \ ... \)\ + x\^n\/\(n!\)\ + \ ... \)]], "\nAt the risk of driving you out of the lab, you are asked to look at this \ with a critical eye. To begin with, nobody has ever succeeded in giving a \ totally precise value for ", Cell[BoxData[ \(e\)]], " itself. This means that if you are going to be totally precise, then you \ have to admit that you are not quite certain of what ", Cell[BoxData[ \(e\^x\)]], " actually means. \nThis problem is compounded by the fact that even if you \ have a totally precise value for ", Cell[BoxData[ \(e\)]], ", then computation of numbers such as ", Cell[BoxData[ \(e\^\[Pi]\)]], " becomes a problem. Raising a number to the ", Cell[BoxData[ \(\[Pi]\^th\)]], " power does not have the same physical meaning that raising a number to \ the ", Cell[BoxData[ \(3\^rd\)]], " power or raising it to the ", Cell[BoxData[ \(\((1\/4)\)\^th\)]], " power by taking its fourth root. In advanced mathematics, these problems \ are cleanly circumvented. \nIn advanced mathematics, ", Cell[BoxData[ \(e\^x\)]], " is defined via the power series\n ", Cell[BoxData[ \(\(1\ + \ x\ + x\^2\/2 + x\^3\/\(3!\) + x\^4\/\(4!\)\ + \ ... \)\ + x\^n\/\(n!\) + ... \)]], "\nYou know this power series converges for ", Cell[BoxData[ \(\(\(-\[Infinity]\)\(\ \)\(<\)\(\ \)\(x\)\(\ \)\(<\)\(\ \)\)\)]], Cell[BoxData[ \(\[Infinity]\)]], "; so from the advanced point of view, there is no lack of clarity or \ precision in the definition.\nSimilarly in advanced mathematics, the \ trigonometric functions ", Cell[BoxData[ \(Sin[x]\)]], " and ", Cell[BoxData[ \(Cos[x]\)]], " are defined via the power series\n ", Cell[BoxData[ \(Sin[ x]\ = \ \(\(x\)\(\ \)\(-\)\(\ \ \)\(x\^3\/\(3!\)\)\(+\)\(x\^5\/\(5!\)\)\(-\)\(x\^7\/\(7!\)\)\(\ \)\(+\)\(\ \)\ \)\)]], " ", Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^n\)\)\ x\^\(2\ n + 1\)\/\(\((2\ \ n + 1)\)!\)\)\(\ \)\(+\)\)\ ... \)]], "\n \n ", Cell[BoxData[ \(Cos[ x]\ = \ \(\(1\)\(\ \)\(-\)\(\ \ \)\(x\^2\/\(2!\)\)\(+\)\(x\^4\/\(4!\)\)\(-\)\(x\^6\/\(6!\)\)\(\ \)\(+\)\(\ \)\ \)\)]], Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^n\)\)\ x\^\(2\ n\)\/\(\((2\ \ n)\)!\)\)\(\ \)\(+\)\)\ ... \)]], "\nThese precise definitions allow you to throw away the compass, \ protractor, and ruler that are usually associated with the measurements \ associated with the trigonometric functions.\nThese two definitions also make \ the mathematics easier too. " }], "Text"], Cell["G.9.a)", "Subsubsection"], Cell[TextData[{ "Give a new derivation of the formula\n ", Cell[BoxData[ \(D[e\^x, \ x]\ = \ e\^x\)]], "\nby differentiating the power series\n ", Cell[BoxData[ \(\(1\ + \ x\ + \ x\^2\/2 + x\^3\/\(3!\) + x\^4\/\(4!\)\ + \ ... \)\ + \ x\^k\/\(k!\)\ + \ ... \)]], "\nand seeing what you get." }], "Text"], Cell["G.9.b)", "Subsubsection"], Cell[TextData[{ "Give a new derivation of the formula\n ", Cell[BoxData[ \(D[Sin[x], \ x]\ = \ Cos[x]\)]], "\nby differentiating the power series\n ", Cell[BoxData[ \(\(\(x\)\(\ \)\(-\)\(\ \)\(x\^3\/\(3!\)\)\(+\)\(x\^5\/\(5!\)\)\(-\)\(x\ \^7\/\(7!\)\)\(\ \)\(+\)\(\ \)\)\)]], " ", Cell[BoxData[ \(\(\(\(\(...\)\(\ \)\(+\ \((\(-1\))\)\^n\)\)\ x\^\(2\ n + 1\)\/\(\((2\ \ n + 1)\)!\)\)\(\ \)\(+\)\)\ ... \)]], "\nand seeing what you get." }], "Text"] }, Closed]], Cell["Literacy Problems", "Subsubsection"], Cell["L.6)", "Subsubsection"], Cell[TextData[{ StyleBox["Find convergence intervals of the following power series.\nUse \ the Power Series Convergence Principle directly, use the ratio test, or use \ any other method you like.\n\[Rule] ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.500496]], Cell[BoxData[ \(1\ + \ x\ + \ x\^2 + \ x\^3 + \ x\^4 + \ x\^5 + \ \(\(.\)\(\ \)\(.\)\)\ . \ \(+\ x\^k\)\)]] }], "Text"], Cell[TextData[{ StyleBox["\[Rule] ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.500496]], Cell[BoxData[ \(1 + \ 2\ x\ + \ 4\ x\^2 + \ 8\ x\^3 + \ 16\ x\^4 + 32\ x\^5 + \ \(\(.\)\(\ \)\(.\)\)\ . \ \(+\ 2\^k\)\ x\^k\)]] }], "Text"], Cell[TextData[{ StyleBox["\[Rule] ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.500496]], Cell[BoxData[ \(1 + x\/5 + x\^2\/5\^2 + x\^3\/5\^3 + x\^4\/5\^4 + \ \(\(.\)\(\ \)\(.\)\)\ . \ \(+\(x\^k\/5\^k\)\)\)]] }], "Text"], Cell[TextData[{ StyleBox["\[Rule] ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 0.500496]], Cell[BoxData[ \(1 + x + x\^2\/2 + x\^3\/3 + x\^4\/4 + \ \(\(.\)\(\ \)\(.\)\)\ . \ \(+\(x\^k\/k\)\)\)]] }], "Text"], Cell[TextData[{ "\[Rule] ", Cell[BoxData[ \(1 + x + x\^2\/\@2 + x\^3\/3\^\(1/3\) + x\^4\/4\^\(1/4\) + \ \(\(.\)\(\ \)\(.\)\)\ . \(+\ \(x\^k\/k\^\((1/k)\ \)\)\)\)]], "\n\nHeavy Tip: ", Cell[BoxData[ \(k\^\((1/k)\) > 1\)]], " for ", Cell[BoxData[ \(k > 1\)]] }], "Text"], Cell["L.8)", "Subsubsection"], Cell[TextData[{ "Suppose ", Cell[BoxData[ \(f[x] = 1 + x + x\^2\/2\^2 + x\^3\/3\^2 + \ \(\(.\)\(\ \)\(.\)\)\ . \(+\ \(x\^k\/k\^2\)\)\)]], "\n\[Rule] What is the value of ", Cell[BoxData[ \(f[0]\)]], "?\n\[Rule] What is the value of ", Cell[BoxData[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]]], "?\n\[Rule] What is the value of ", Cell[BoxData[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "0", "]"}]]], "?\n\[Rule] What is the value of ", Cell[BoxData[ RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", 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Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], ShowCellBracket->False, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {0, 0}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Times", FontSize->14], Cell[StyleData["Subtitle", "Presentation"], CellFrame->False, CellMargins->{{24, Inherited}, {6, Inherited}}], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}], Cell[StyleData["Subtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellFrame->True, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {6, Inherited}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Times", FontSize->24, FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Subsubtitle", "Presentation"], CellFrame->True, CellMargins->{{24, Inherited}, {6, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]], Cell[StyleData["Subsubtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{22, Inherited}, {Inherited, 20}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Times", FontSize->16], Cell[StyleData["Section", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 20}}, LineSpacing->{1.5, 0}, FontSize->18], Cell[StyleData["Section", "Printout"], CellMargins->{{14, Inherited}, {2, 10}}], Cell[StyleData["Section", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 10}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{19, Inherited}, {Inherited, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0.392187, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 15}}], Cell[StyleData["Subsection", "Printout"], CellMargins->{{14, Inherited}, {2, 5}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsection", "TwoColumn"], CellFrame->{{0, 0}, {0, 1}}, CellMargins->{{14, Inherited}, {2, 10}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, CellMargins->{{18, Inherited}, {Inherited, 12}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{14, Inherited}, {2, 3}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsubsection", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 3}}, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceText"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->10, FontWeight->"Plain"], Cell[StyleData["PrefaceText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["PrefaceText", "Printout"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None], Cell[StyleData["PrefaceText", "TwoColumn"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[StyleData["PrefaceHyperlink"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[CellGroupData[{ Cell[StyleData["Text"], ShowClosedCellArea->True, CellMargins->{{15, 10}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 0.500008]], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Text", "Printout"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontColor->GrayLevel[0], Background->None], Cell[StyleData["Text", "TwoColumn"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontSize->14, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], ShowClosedCellArea->True, CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["SmallText", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True], Cell[StyleData["SmallText", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {Inherited, Inherited}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Printout"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->9], Cell[StyleData["Input", "TwoColumn"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1.5, 0}, FormatType->StandardForm, FontFamily->"Courier", FontSize->14, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Output", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]], Cell[StyleData["Message", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9], Cell[StyleData["Print", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Info"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Info", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Info", "Printout"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["Info", "TwoColumn"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, AnimationDisplayTime->0.2, StyleMenuListing->None, FontSize->14], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{30, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{45, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, FontSize->9], Cell[StyleData["Graphics", "TwoColumn"], CellMargins->{{20, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontColor->GrayLevel[1]], Cell[StyleData["CellLabel", "TwoColumn"], FontColor->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceTitle"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0, 0.392187, 0]], Cell[StyleData["PrefaceTitle", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->38, FontColor->GrayLevel[1], Background->RGBColor[0.596078, 0.65098, 0.0196078]], Cell[StyleData["PrefaceTitle", "Printout"], CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["PrefaceTitle", "TwoColumn"], CellFrame->{{1, 1}, {0, 5}}, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Body Text and Index", "Section"], Cell[CellGroupData[{ Cell[StyleData["Accident"], CellFrame->3, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Accident", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->36], Cell[StyleData["Accident", "Printout"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Accident", "TwoColumn"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ContentsText"], CellMargins->{{50, 10}, {5, 5}}, FontFamily->"Times", FontSize->16], Cell[StyleData["ContentsText", "Presentation"]], Cell[StyleData["ContentsText", "Printout"], FontColor->GrayLevel[0], Background->None], Cell[StyleData["ContentsText", "TwoColumn"], FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special1"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Special1", "Presentation"], FontSize->16], Cell[StyleData["Special1", "Printout"], FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Special1", "TwoColumn"], FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special2"], CellMargins->{{6, 0}, {0, 0}}, CellGroupingRules->{"SectionGrouping", 40}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special2", "Presentation"], FontSize->12], Cell[StyleData["Special2", "Printout"], FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Special2", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special3"], CellDingbat->"\[GraySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special3", "Presentation"]], Cell[StyleData["Special3", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special3", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special4"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special4", "Presentation"], FontSize->12], Cell[StyleData["Special4", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special4", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special5"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special5", "Presentation"], FontSize->12], Cell[StyleData["Special5", "Printout"]], Cell[StyleData["Special5", "TwoColumn"]] }, Closed]], Cell[StyleData["IndexEntry"], ShowCellBracket->False, CellMargins->{{15, 5}, {0, 5}}, PageBreakBelow->False, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16], Cell[StyleData["IndexSubEntry"], ShowCellBracket->False, CellMargins->{{45, 5}, {0, 0}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["Styles for License Agreement", "Section"], Cell[CellGroupData[{ Cell[StyleData["LicenseHeading"], ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{24, 24}, {-1, 2}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.4, 0.300008, 0.6]], Cell[StyleData["LicenseHeading", "Presentation"], FontSize->12], Cell[StyleData["LicenseHeading", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseHeading", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LicenseText"], CellFrame->True, ShowCellBracket->False, CellMargins->{{24, 24}, {5, -1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, Background->RGBColor[1, 0.537743, 0.509071]], Cell[StyleData["LicenseText", "Presentation"], FontSize->18], Cell[StyleData["LicenseText", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseText", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CounterIncrements->"NumberedEquation"], Cell[StyleData["NumberedEquation", "Presentation"]], Cell[StyleData["NumberedEquation", "Printout"]], Cell[StyleData["NumberedEquation", "TwoColumn"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedFigure", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{6, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedTable", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. 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