Investigate the Convergence of Infinite Series
Mathematica 7 provides dedicated functionality for determining the convergence conditions of infinite series.
 In[1]:= ```Text@TraditionalForm@ Grid[{{HoldForm[\!\(TraditionalForm\`SumConvergence[ \*FractionBox[\(1\), SuperscriptBox[\(n\), \(2\)]], n]\)], \!\(TraditionalForm\`SumConvergence[ \*FractionBox[\(1\), SuperscriptBox[\(n\), \(2\)]], n]\), DiscretePlot[\!\(TraditionalForm\` \*UnderoverscriptBox[\(\[Sum]\), \(n\), \(m\)] \*FractionBox[\(1\), SuperscriptBox[\(n\), \(2\)]]\), {m, 20}, ImageSize -> 100, Ticks -> None]}, {HoldForm[\!\(TraditionalForm\`SumConvergence[n, n]\)], \!\(TraditionalForm\`SumConvergence[n, n]\), DiscretePlot[\!\(TraditionalForm\` \*UnderoverscriptBox[\(\[Sum]\), \(n\), \(m\)]n\), {m, 20}, ImageSize -> 100, Ticks -> None]}, {HoldForm[\!\(TraditionalForm\`SumConvergence[\* FormBox[ SuperscriptBox["x", "n"], TraditionalForm], n]\)], \!\(TraditionalForm\`SumConvergence[\* FormBox[ SuperscriptBox["x", "n"], TraditionalForm], n] // TraditionalForm\), Row[Table[DiscretePlot[Sum[\!\(TraditionalForm\` \*SuperscriptBox[\(x\), \(n\)]\), {n, m}], {m, 15}, ImageSize -> 75, Ticks -> None], {x, \!\(TraditionalForm\`{ \*FractionBox[\(1\), \(3\)], 1, \*FractionBox[\(4\), \(3\)]}\)}]]}, \ {HoldForm[\!\(TraditionalForm\`SumConvergence[\* FormBox[ FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"(", RowBox[{"n", "+", "1"}], ")"}]], "n"], TraditionalForm], n]\)], \!\(TraditionalForm\`SumConvergence[\* FormBox[ RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], "n"]}], TraditionalForm], n]\), DiscretePlot[Sum[\!\(TraditionalForm\` \*FractionBox[ SuperscriptBox[\((\(-1\))\), \((n + 1)\)], \(n\)]\), {n, m}], {m, 30}, ImageSize -> 100, Ticks -> None]}, {HoldForm[\!\(TraditionalForm\`SumConvergence[\* FormBox[ FractionBox[ RowBox[{ TemplateBox[{"n","5"}, "Mod"], " ", RowBox[{"sin", "(", RowBox[{"4", " ", "n"}], ")"}]}], "n"], TraditionalForm], n]\)], \!\(TraditionalForm\`SumConvergence[\* FormBox[ FractionBox[ RowBox[{ TemplateBox[{"n","5"}, "Mod"], " ", RowBox[{"sin", "(", RowBox[{"4", " ", "n"}], ")"}]}], "n"], TraditionalForm], n]\), DiscretePlot[Sum[\!\(TraditionalForm\`\* FractionBox[ RowBox[{ TemplateBox[{"n","3"}, "Mod"], " ", RowBox[{"sin", "(", RowBox[{"4", " ", "n"}], ")"}]}], "n"]\), {n, m}], {m, 20, 180, 2}, ImageSize -> 100, Ticks -> None]}, {HoldForm[ SumConvergence[n Boole[n^2 < 1000], n]], \!\(TraditionalForm\`SumConvergence[\* FormBox[ RowBox[{"n", " ", RowBox[{"Boole", "[", RowBox[{ SuperscriptBox["n", "2"], "<", "1000"}], "]"}]}], TraditionalForm], n]\), DiscretePlot[\!\(TraditionalForm\` \*UnderoverscriptBox[\(\[Sum]\), \(n\), \(m\)]n\ Boole[ \*SuperscriptBox[\(n\), \(2\)] < 1000]\), {m, 1, 80, 2}, ImageSize -> 100, Ticks -> None]}}, Dividers -> All, Background -> LightYellow, Alignment -> {{Center, Center, Center}, Baseline}, Spacings -> {1, 1.6}]```
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