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\(TraditionalForm\`s\_354 = 4.8\)]], " to ", Cell[BoxData[ \(TraditionalForm\`s\_355 = 29.4\)]], "!" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[355/113]\)], "Input"], Cell[BoxData[ \(3.14159292035398252`\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`355\ \[TildeEqual] \ 113 \[Pi]\)]], ", ", Cell[BoxData[ \(TraditionalForm \`sin(355) \[TildeEqual] \ sin(113 \[Pi])\ = \ 0\)]], " and the 355th term of the series suddenly becomes very large compared to \ nearby terms:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(1. \/\(354\^3\ Sin[354]\^2\)\), \(1. \/\(355\^3\ Sin[355]\^2\)\), \(1. \/\(356\^3\ Sin[356]\^2\)\)}], "Input"], Cell[BoxData[ \(3.18367506175675929`*^-8\)], "Output"], Cell[BoxData[ \(24.5981812207061034`\)], "Output"], Cell[BoxData[ \(3.13007615298842534`*^-8\)], "Output"] }, Open ]], Cell[TextData[{ "We should expect to see a jump in ", Cell[BoxData[ \(TraditionalForm\`s\_n\)]], " whenever ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is close enough to a multiple of \[Pi] to make ", Cell[BoxData[ \(TraditionalForm\`\(n\^3\) \(sin(n)\)\)]], "a small number." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[If\ [ FractionalPart[N[\ n\ \[Pi]\ ]] < .001 || FractionalPart[N[\ n\ \[Pi]\ ]] > .999, Print[\ N[n\ \[Pi]]]], {n, 100, 500}]\)], "Input"], Cell[BoxData[ \(354.999969855646613`\)], "Print"], Cell[BoxData[ \(709.999939711293226`\)], "Print"], Cell[BoxData[ \(1064.99990956693979`\)], "Print"], Cell[BoxData[ \(1419.99987942258654`\)], "Print"] }, Open ]], Cell[TextData[{ "Jumps should occur when ", Cell[BoxData[ \(TraditionalForm\`n = 710\)]], ", ", Cell[BoxData[ \(TraditionalForm\`n = 1065\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`n = 1420\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(s[708]\), \(s[709]\), \(s[710]\), \(s[711]\)}], "Input"], Cell[BoxData[ \(29.4060425701563801`\)], "Output"], Cell[BoxData[ \(29.4060425741193132`\)], "Output"], Cell[BoxData[ \(30.174735737964875`\)], "Output"], Cell[BoxData[ \(30.1747357418938477`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(s[1063]\), \(s[1064]\), \(s[1065]\), \(s[1066]\)}], "Input"], Cell[BoxData[ \(30.1748051114131898`\)], "Output"], Cell[BoxData[ \(30.1748051125857852`\)], "Output"], Cell[BoxData[ \(30.2760321960438538`\)], "Output"], Cell[BoxData[ \(30.276032197209588`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(s[1418]\), \(s[1419]\), \(s[1420]\), \(s[1421]\)}], "Input"], Cell[BoxData[ \(30.2760557193852086`\)], "Output"], Cell[BoxData[ \(30.2760557198795643`\)], "Output"], Cell[BoxData[ \(30.3000773813370516`\)], "Output"], Cell[BoxData[ \(30.3000773818291735`\)], "Output"] }, Open ]], Cell[TextData[{ "The jumps are small, but very noticible compared to how little the nearby \ partial sums are changing. Such jumps make it difficult to assess whether the \ series will ultimately converge. We do not have an easy way to estimate how \ small ", Cell[BoxData[ \(TraditionalForm\`sin(n)\)]], " may become (it is never zero of course), and therefore, no easy way to \ estimate how large ", Cell[BoxData[ \(TraditionalForm\`1\/\(\(n\^3\) \(\(sin\^2\)(n)\)\)\)]], " may become. 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