(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 4756, 176]*) (*NotebookOutlinePosition[ 5415, 200]*) (* CellTagsIndexPosition[ 5371, 196]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Functions of a Complex Variable/Series and Convergence", "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell["Series of Constants", "Subtitle"], Cell[TextData[{ "Sean Mauch\nsean@its.caltech.edu\n", ButtonBox["http://www.its.caltech.edu/~sean", ButtonData:>{ URL[ "http://www.its.caltech.edu/~sean"], None}, ButtonStyle->"Hyperlink"], "\n4/19/00" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[n, x]\)], "Input"], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ \(\[Sum]\_\(n = 2\)\%\[Infinity] 1\/\(n\ Log[n]\)\)]]], "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will return the input. It does not say that the sum is divergent." }], "Text"], Cell[BoxData[ \(\[Sum]\_\(n = 2\)\%\[Infinity] 1\/\(n\ Log[n]\)\)], "Input"], Cell["\<\ Trying to sum it numerically makes us suspect it is \ divergent.\ \>", "Text"], Cell[BoxData[ \(NSum[1\/\(n\ Log[n]\), {n, 2, \[Infinity]}]\)], "Input"], Cell["\<\ Since the associated integral diverges, the sum is divergent.\ \>", \ "Text"], Cell[BoxData[ \(\[Integral]\_2\%\[Infinity]\( 1\/\(x\ Log[x]\)\) \[DifferentialD]x\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity]\((\(-1\))\)\^\(n + 1\)\/n\)]]], "Section"], Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity]\((\(-1\))\)\^\(n + 1\)\/n\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity] 1\/n\)]]], "Section"], Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity] 1\/n\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity]\( n!\)\/n\^n\)]]], "Section"], Cell[TextData[{ "Since this does not sum to a simple expression, ", StyleBox["Mathematica", FontSlant->"Italic"], " returns the input." }], "Text"], Cell[BoxData[ \(\[Sum]\_\(n = 1\)\%\[Infinity]\( n!\)\/n\^n\)], "Input"], Cell[TextData[{ "Since ", StyleBox["Mathematica", FontSlant->"Italic"], " does not complain when doing the numerical sum, we suspect that the sum \ converges." }], "Text"], Cell[BoxData[ \(NSum[\(n!\)\/n\^n, {n, 1, \[Infinity]}]\)], "Input"], Cell["We apply the ratio test.", "Text"], Cell[BoxData[ \(Limit[\(\(\((n + 1)\)!\)/\((n + 1)\)\^\(n + 1\)\)\/\(\(n!\)/n\^n\), n \[Rule] \[Infinity]]\)], "Input"], Cell[TextData[{ "That choked. However, if we simplify the expression, ", StyleBox["Mathematica", FontSlant->"Italic"], " can evaluate the limit." }], "Text"], Cell[BoxData[ \(\(\(\((n + 1)\)!\)/\((n + 1)\)\^\(n + 1\)\)\/\(\(n!\)/n\^n\) // FullSimplify\)], "Input"], Cell[BoxData[ \(Limit[%, n \[Rule] \[Infinity]]\)], "Input"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`1\/\[ExponentialE] < 1\)]], ", so the sum converges by the ratio test." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for X", ScreenRectangle->{{0, 1152}, {0, 864}}, WindowSize->{520, 600}, WindowMargins->{{Automatic, 18}, {Automatic, 0}}, Magnification->1.25 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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