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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[ 293330, 7677]*) (*NotebookOutlinePosition[ 294000, 7701]*) (* CellTagsIndexPosition[ 293956, 7697]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Examples for Section 8.10 Convegence of Taylor Series\ \>", "Subsection"], Cell[CellGroupData[{ Cell["p.687", "Subsection"], Cell[CellGroupData[{ Cell["1.", "Subsubsection"], Cell[TextData[{ "Plug ", Cell[BoxData[ \(TraditionalForm\`u = \(-5\) x\)]], " into the series ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(e\^u = 1 + u + \(1\/\(2!\)\) u\^2 + \ \(1\/\(3!\)\) u\^3 + \ ... \), "TraditionalForm"], "+", \(\(1\/\(n!\)\) u\^n\), "+"}], " ", "..."}], TraditionalForm]]], " to get", "\n", Cell[BoxData[ \(TraditionalForm\`\(e\^\(\(-5\) x\) = \)\)]], Cell[BoxData[ \(TraditionalForm \`\(\(\(1 + \((\(-5\) x)\) + \(1\/\(2!\)\) \((\(-5\) x)\)\^2 + \ \(1\/\(3!\)\) \((\(-5\) x)\)\^3 + \ ... \) + \(1\/\(n!\)\) \((\(-5\) x)\)\^n + \ ... = \)\ \)\)]], Cell[BoxData[ \(TraditionalForm \`\(1 - 5 x + \(5\^2\/\(2!\)\) x\^2 + \ \(5\^3\/\(3!\)\) x\^3 + \ ... \) + \(\(\(\((\(-1\))\)\^n\) 5\^n\)\/\(n!\)\) x\^n + \ ... \)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[E\^\(\(-5\) x\), {x, 0, 4}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "-", \(5\ x\), "+", \(\(25\ x\^2\)\/2\), "-", \(\(125\ x\^3\)\/6\), "+", \(\(625\ x\^4\)\/24\), "+", InterpretationBox[\(O[x]\^5\), SeriesData[ x, 0, {}, 0, 5, 1]]}], SeriesData[ x, 0, {1, -5, Rational[ 25, 2], Rational[ -125, 6], Rational[ 625, 24]}, 0, 5, 1]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["5. ", "Subsubsection"], Cell[TextData[{ "Plug ", Cell[BoxData[ \(TraditionalForm\`u = \@x\)]], " into the series ", Cell[BoxData[ \(TraditionalForm \`cos(u)\ = \ \(1\ - \ \(1\/\(2!\)\) u\^2 + \ \(1\/\(4!\)\) u\^4 + \ ... \) + \(\((\(-1\))\)\^n\/\(\((2 n)\)!\)\) u\^\(2 n\) + \ ... \)]], " to get ", Cell[BoxData[ \(TraditionalForm \`cos(\@x)\ = \ \(\(1\ - \ \(1\/\(2!\)\) \((\@x)\)\^2 + \ \(1\/\(4!\)\) \((\@x)\)\^4 + \ ... \) + \(\((\(-1\))\)\^n\/\(\((2 n)\)!\)\) \((\@x)\)\^\(2 n\) + \ ... = \ \(1\ - \ \(1\/\(2!\)\) x + \ \(1\/\(4!\)\) x\^2 + \ ... \) + \(\((\(-1\))\)\^n\/\(\((2 n)\)!\)\) x\^n + \ ... \)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[Cos[\@x], {x, 0, 4}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "-", \(x\/2\), "+", \(x\^2\/24\), "-", \(x\^3\/720\), "+", \(x\^4\/40320\), "+", InterpretationBox[\(O[x]\^5\), SeriesData[ x, 0, {}, 0, 10, 2]]}], SeriesData[ x, 0, {1, 0, Rational[ -1, 2], 0, Rational[ 1, 24], 0, Rational[ -1, 720], 0, Rational[ 1, 40320]}, 0, 10, 2]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["15.", "Subsubsection"], Cell[TextData[{ "Plug ", Cell[BoxData[ \(TraditionalForm\`u = 2 x\)]], " into the formula for geometric series ", Cell[BoxData[ \(TraditionalForm \`1\/\(1 - u\) = \(1 + u + u\^2 + ... \) + u\^n + \ ... \)]], " to get:\n", Cell[BoxData[ FormBox[ RowBox[{\(x\^2\/\(1 - 2 x\)\), "=", RowBox[{\(\(x\^2\)(1\/\(1 - 2 x\))\), " ", "=", " ", RowBox[{ \(\(x\^2\)( \(1\ + \ \((2 x)\)\ + \ \((2 x)\)\^2 + \ ... \)\ + \ \((2 x)\)\^n + \ ... )\), " ", "=", " ", FormBox[\(\(x\^2 + \ 2 x\^3 + \ \(2\^2\) x\^4 + \ ... \)\ + \ \(2\^n\) x\^\(n + 2\) + \ ... \), "TraditionalForm"]}]}]}], TraditionalForm]]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[x\^2\/\(1 - 2 x\), {x, 0, 4}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(x\^2\), "+", \(2\ x\^3\), "+", \(4\ x\^4\), "+", InterpretationBox[\(O[x]\^5\), SeriesData[ x, 0, {}, 2, 5, 1]]}], SeriesData[ x, 0, {1, 2, 4}, 2, 5, 1]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["17.", "Subsubsection"], Cell[TextData[{ "First observe that ", Cell[BoxData[ \(TraditionalForm \`\(d\/\(d\ x\)\) \((1\/\(1 - x\))\) = 1\/\((1 - x)\)\^2\)]], " [(for problem 18, you also need ", Cell[BoxData[ \(TraditionalForm \`\(d\/\(d\ x\)\) \((1\/\((1 - x)\)\^2)\) = 2\/\((1 - x)\)\^3\)]], "], so ", Cell[BoxData[ \(TraditionalForm \`1\/\((1 - x)\)\^2 = \(\(d\/\(d\ x\)\) \((1\/\(1 - x\))\) = \(\(d\/\(d\ x\)\) \((\(1 + x + x\^2 + x\^3 + ... \) + x\^n + ... )\) = \(0 + 1 + 2 x + 3 x\^2 + ... \) + n\ x\^\(n - 1\) + ... \)\)\)]], "\nYou could also try multiplying the series, but that is a little harder: \ ", Cell[BoxData[ \(TraditionalForm \`1\/\((1 - x)\)\^2 = \(\((\(1 + x + x\^2 + ... \) + x\^n + ... )\) \((\(1 + x + x\^2 + ... \) + x\^n + ... )\) = \(1 + 2 x + 3 x\^2 + ... \) + \((n\ + 1)\) x\^n + ... \)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[1\/\((1 - x)\)\^2, {x, 0, 4}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(2\ x\), "+", \(3\ x\^2\), "+", \(4\ x\^3\), "+", \(5\ x\^4\), "+", InterpretationBox[\(O[x]\^5\), SeriesData[ x, 0, {}, 0, 5, 1]]}], SeriesData[ x, 0, {1, 2, 3, 4, 5}, 0, 5, 1]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["19.", "Subsubsection"], Cell[TextData[{ "Fir approximately what values of ", Cell[BoxData[ \(TraditionalForm\`x\)]], " can you replace ", Cell[BoxData[ \(TraditionalForm\`sin(x)\)]], " by ", Cell[BoxData[ \(TraditionalForm\`x\ - \ \((x\^3/6)\)\)]], " with an error of magnitude no greater than ", Cell[BoxData[ \(TraditionalForm\`5\[Times]10\^\(-4\)\)]], "?\n\n", StyleBox["Solution.", FontSlant->"Italic"], " The error of approximation is less than ", Cell[BoxData[ \(TraditionalForm \`\( | \(R\_3\)(x) | \) = \( | \((\(\(\(\(sin'\)'\)'\)' \((c)\)\)\/\(4!\))\) x\^4 | \( \[LessEqual] \ \(1\/24\) x\^4\)\)\)]], ". 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" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[\@\(1 + x\), {x, 3}]\)], "Input"], Cell[BoxData[ \(3\/\(8\ \((1 + x)\)\^\(5/2\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm \`3\/\(8\ \((1 + c)\)\^\(5/2\)\) < \((3\/\(8 \((1 - 0.01)\)\^\(5/2\)\))\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\( = 0.384542\)\)]], " for ", Cell[BoxData[ \(TraditionalForm\`\(\(-0.01\) < x < 0.01, \)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", FormBox[\(3\/\(8 \((1 - 0.01)\)\^\(5/2\)\)\), "TraditionalForm"]}]], "Input"], Cell[BoxData[ \(0.384541557720849436`\)], "Output"] }, Open ]], Cell[TextData[{ "we see that we can estimate, using ", Cell[BoxData[ \(TraditionalForm\`\( | x | \( < 0.01\)\)\)]], " , ", Cell[BoxData[ \(TraditionalForm \`\( | \(R\_2\)(x) | \ \( < \ 0.384542\)\ | x \( | \^3\)\( < \ \ 0.384542\ \((0.1)\)\^3\)\) = 0.000384542\)]], " and the error is never greater than ", Cell[BoxData[ 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