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The series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/\@n\)]], " is a ", Cell[BoxData[ \(TraditionalForm\`p\)]], "-series with ", Cell[BoxData[ \(TraditionalForm\`p < 1\)]], ", so it diverges. Hence, the original series diverges too by the \ Comparison Test." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["3.", "Subsubsection"], Cell[TextData[{ "Compare the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\(\( sin\^2\)(n)\)\/2\^n\)]], " with ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/2\^n\)]], " since ", Cell[BoxData[ \(TraditionalForm\`\(\(sin\^2\)(n)\)\/2\^n < 1\/2\^n\)]], ". The latter series converges as a geometric series, so the former series \ converges too. \nThe function ", StyleBox["NSum", "Input"], " can be used to compute a numerical approximation for the series:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NSum[Sin[n]\/2\^n, {n, 1, \[Infinity]}]\)], "Input"], Cell[BoxData[ \(0.592837622028730137`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["5.", "Subsubsection"], Cell[TextData[{ "The series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\( 2 n\)\/\(3 n - 1\)\)]], " does not converge because the terms do not approach 0: ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)\(2 n\)\/\(3 n - 1\) = 2\/3 \[NotEqual] 0\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["11.", "Subsubsection"], Cell[TextData[{ "For the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity]\((ln(n))\)\^2\/n\^3\)]], ", use the Limit Comparison Test with ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^2\)]], ". The latter series converges as a ", Cell[BoxData[ \(TraditionalForm\`p\)]], "-series, and ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)a\_n\/b\_n = \(lim\_\(n \[Rule] \[Infinity]\)\(( \(\((ln(n))\)\^2\/n\^3\)/\(1\/n\^2\))\) = \(lim\_\(n \[Rule] \[Infinity]\)\((ln(n))\)\^2\/n = 0\)\)\)]], " by L'Hopital's Rule. Therefore the original series converges." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\(\((Log[n])\)\^2\/n\^3\)/\(1\/n\^2\), n \[Rule] \[Infinity]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NSum[\((Log[n])\)\^2\/n\^3, {n, 1, \[Infinity]}]\)], "Input"], Cell[BoxData[ \(0.23974691725596533`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["19.", "Subsubsection"], Cell[TextData[{ "Compare the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 2\)\%\[Infinity] 1\/\(n \@\( n\^2 - 1\)\)\)]], " with ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 2\)\%\[Infinity] 1\/n\^2\)]], " using the limit comparison test: ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)\(( \(1\/\(n \@\( n\^2 - 1\)\)\)/\(1\/n\^2\))\) = \(lim\_\(n \[Rule] \[Infinity]\)1\/\@\(1 - 1\/n\^2\) = 1\)\)]], " . The latter series is a ", Cell[BoxData[ \(TraditionalForm\`p\)]], "-series and converges since ", Cell[BoxData[ \(TraditionalForm\`p = 2 > 1\)]], ", so the former series converges as well." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\(1\/\(n \@\( n\^2 - 1\)\)\)/\(1\/n\^2\), n \[Rule] \[Infinity]] \)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NSum[1\/\(n \@\( n\^2 - 1\)\), {n, 2, \[Infinity]}]\)], "Input"], Cell[BoxData[ \(0.69422401991054139`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["25.", "Subsubsection"], Cell[TextData[{ "The series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] sin(1\/n)\)]], " diverges because we can compare it to the harmonic series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\)]], " which diverges: ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)\(sin(1\/n)\)/\(1\/n\) = 1\)]], " (by L'Hopital's Rule)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[Sin[1\/n]/\(1\/n\), \ n \[Rule] \[Infinity]]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["41. 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