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The original series is also a \ geometric series: ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\(\((\(-1\))\)\^\(n + 1\)\) \((0.1)\)\^n\ = \ \(\(-\(\[Sum]\+\(n = 1\)\%\[Infinity]\((\(-0.1\))\)\^n\)\) = \(\(-\(\((\(-0.1\))\)\/\(1 - \((\(-0.1\))\)\)\)\) = \(1\/11 = 0.090909 ... \)\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSum[\(\((\(-1\))\)\^\(n + 1\)\) \((0.1)\)\^n, {n, 1, \[Infinity]}]\)], "Input"], Cell[BoxData[ \(0.0909090909090909171`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["21.", "Subsubsection"], Cell[TextData[{ "The series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\(\((\(-1\))\)\^\(n + 1\)\) \((\(1 + n\)\/n\^2)\)\)]], " is convergent because it is an alternating series whose terms are \ decreasing and approach 0: ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)\(1 + n\)\/n\^2 = 0\)]], ". The series is not absolutely convergent because the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\((\(1 + n\)\/n\^2)\)\)]], " diverges: ", Cell[BoxData[ \(TraditionalForm\`\(1 + n\)\/n\^2 = 1\/n\^2 + 1\/n > 1\/n\)]], " and the harmonic series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\)]], " diverges. Therefore, the original series is conditionally convergent." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSum[ \(\((\(-1\))\)\^\(n + 1\)\) \((\(1 + n\)\/n\^2)\), {n, 1, \[Infinity]}] \)], "Input"], Cell[BoxData[ \(1.51561421398405737`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["45.", "Subsubsection"], Cell[TextData[{ "Estimate the magnitude of the error in using the sum of the first four \ terms to approximate the sum of the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\) 1\/n\)]], ".\n\nSince the series is alternating, the error is less than the absolute \ value of the 5th term: ", Cell[BoxData[ \(TraditionalForm \`\( | error | \( \[LessEqual] \((1\/5)\)\)\) = 0.2\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(1 - 1\/2 + 1\/3 - 1\/4 // N\)], "Input"], Cell[BoxData[ \(0.583333333333333392`\)], "Output"] }, Open ]], Cell[TextData[{ "The series actually sums to ", Cell[BoxData[ \(TraditionalForm\`ln(2)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Log[2] // N\)], "Input"], Cell[BoxData[ \(0.693147180559945308`\)], "Output"] }, Open ]], Cell["The error in using the firts four terms is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((1 - 1\/2 + 1\/3 - 1\/4)\) - Log[2] // N\)], "Input"], Cell[BoxData[ \(\(-0.109813847226611915`\)\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, ScreenStyleEnvironment->"Working", WindowSize->{542, 600}, WindowMargins->{{151, Automatic}, {12, Automatic}} ] (*********************************************************************** Cached data follows. 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