(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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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763.375}, {471.375, 0}} -> {-7.35721*^-9, 2.71829, \ 1.43911*^-10, 7.83178*^-10}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "\nBut ", StyleBox["Mathematica", FontSlant->"Italic"], " knows the value of this limit:" }], "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell["Some limits", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`lim\+\(x \[Rule] \[Infinity]\)\[ThinSpace]\((1 + 1\/x)\ \)\^x\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\)], "Output"] }, Open ]], Cell["\<\ and we can even obtain information about how this limit is \ obtained:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ FractionBox[\(\[PartialD]\((1 + 1\/x)\)\^x\), \(\[PartialD]x\), MultilineFunction->None], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`\((1 + 1\/x)\)\^x\ \((log(1 + 1\/x) - 1\/\(\((1 + 1\/x)\)\ x\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`lim\+\(x \[Rule] \[Infinity]\)\[ThinSpace]%\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ It is a nice asymptote. Inverting near zero we get\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`lim\+\(x \[Rule] 0\)\[ThinSpace]\((x + \ 1)\)\^\(1/x\)\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\)], "Output"] }, Open ]], Cell["Direct computation of the slope near zero is not easy:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ FractionBox[\(\[PartialD]\((x + 1)\)\^\(1/x\)\), \(\[PartialD]x\), MultilineFunction->None], TraditionalForm]], "Input"], Cell[BoxData[ \(\((1 + x)\)\^\(1\/x\)\ \((1\/\(x\ \((1 + x)\)\) - Log[1 + x]\/x\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[%, x -> 0]\)], "Input"], Cell[BoxData[ \(\(-\(\[ExponentialE]\/2\)\)\)], "Output"] }, Open ]], Cell["Here is why:", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ FractionBox[\(\[PartialD]\(log(\((x + 1)\)\^\(1/x\))\)\), \ \(\[PartialD]x\), MultilineFunction->None], TraditionalForm]], "Input"], Cell[BoxData[ \(1\/\(x\ \((1 + x)\)\) - Log[1 + x]\/x\^2\)], "Output"] }, Open ]], Cell["We recognise factors and do the factoring", "Text", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(Apart[%]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(1\/\(x\ \((1 + x)\)\) - Log[1 + x]\/x\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[%, x -> 0]\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\)\)], "Output"] }, Open ]], Cell["Therefore the slope is", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(TraditionalForm\`\(-1\)\/\(2\ \[ExponentialE]\)\)], "Input", Evaluatable->False, CellAutoOverwrite->False], Cell["", "Text", CellAutoOverwrite->False], Cell["\<\ The problem is that we cannot get this information in a plot near \ zero.\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["A simple reasoning", "Section", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell["Taking a long polynomial", "Subsection"], Cell["\<\ Use Mc Laurin development to approximate the previous plots by some \ polynomials. We do not have to go all the way to\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(TraditionalForm\`Series[log(x + 1), {x, 0, 50}]; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(exponent = Normal[%]\)], "Input"], Cell[BoxData[ \(x - x\^2\/2 + x\^3\/3 - x\^4\/4 + x\^5\/5 - x\^6\/6 + x\^7\/7 - x\^8\/8 + x\^9\/9 - x\^10\/10 + x\^11\/11 - x\^12\/12 + x\^13\/13 - x\^14\/14 + x\^15\/15 - x\^16\/16 + x\^17\/17 - x\^18\/18 + x\^19\/19 - x\^20\/20 + x\^21\/21 - x\^22\/22 + x\^23\/23 - x\^24\/24 + x\^25\/25 - x\^26\/26 + x\^27\/27 - x\^28\/28 + x\^29\/29 - x\^30\/30 + x\^31\/31 - x\^32\/32 + x\^33\/33 - x\^34\/34 + x\^35\/35 - x\^36\/36 + x\^37\/37 - x\^38\/38 + x\^39\/39 - x\^40\/40 + x\^41\/41 - x\^42\/42 + x\^43\/43 - x\^44\/44 + x\^45\/45 - x\^46\/46 + x\^47\/47 - x\^48\/48 + x\^49\/49 - x\^50\/50\)], "Output"] }, Open ]], Cell[BoxData[ \(TraditionalForm\`Series[\[ExponentialE]\^\(exponent/x\), {x, 0, 50}]; \)], "Input"], Cell["or we do not have to go even to 20", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(approxim = Normal[Series[%, {x, 0, 20}]]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(\[ExponentialE] - \(\[ExponentialE]\ x\)\/2 + \(11\ \[ExponentialE]\ \ x\^2\)\/24 - \(7\ \[ExponentialE]\ x\^3\)\/16 + \(2447\ \[ExponentialE]\ x\^4\ \)\/5760 - \(959\ \[ExponentialE]\ x\^5\)\/2304 + \(238043\ \[ExponentialE]\ \ x\^6\)\/580608 - \(67223\ \[ExponentialE]\ x\^7\)\/165888 + \(559440199\ \ \[ExponentialE]\ x\^8\)\/1393459200 - \(123377159\ \[ExponentialE]\ \ x\^9\)\/309657600 + \(29128857391\ \[ExponentialE]\ x\^10\)\/73574645760 - \ \(5267725147\ \[ExponentialE]\ x\^11\)\/13377208320 + \(9447595434410813\ \ \[ExponentialE]\ x\^12\)\/24103053950976000 - \(1447646915836493\ \ \[ExponentialE]\ x\^13\)\/3708162146304000 + \(225037938358318573\ \ \[ExponentialE]\ x\^14\)\/578473294823424000 - \(29911565062525361\ \ \[ExponentialE]\ x\^15\)\/77129772643123200 + \(3651003047854884043877\ \ \[ExponentialE]\ x\^16\)\/9440684171518279680000 - \(38950782815463986767\ \ \[ExponentialE]\ x\^17\)\/100969884187361280000 + \ \(104388909491649724435759747\ \[ExponentialE]\ \ x\^18\)\/271211974879377138647040000 - \(843527474424845338522843\ \ \[ExponentialE]\ x\^19\)\/2196048379590098288640000 + \ \(1372557084260440289321615059133\ \[ExponentialE]\ \ x\^20\)\/3579998068407778230140928000000\)], "Output"] }, Open ]], Cell["\<\ which is unreadable and of no intrest! In fact, a quadratic approximation is sufficient for our purposes:\ \>", \ "Text", CellAutoOverwrite->False] }, Open ]], Cell[CellGroupData[{ Cell["A smaller degree polynomial is sufficient", "Subsection"], Cell["\<\ It is sufficient to write an approxim of second order,which we \ could do even by hand :\ \>", "Text", CellAutoOverwrite->False], Cell[BoxData[ \(TraditionalForm\`Series[log(x + 1), {x, 0, 2}]; \)], "Input", CellAutoOverwrite->False], Cell[CellGroupData[{ Cell[BoxData[ \(exponent2 = Normal[%]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ \(x - x\^2\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`Series[\[ExponentialE]\^\(exponent2/x\), {x, 0, 1}]\)], "Input", CellAutoOverwrite->False], Cell[BoxData[ InterpretationBox[ RowBox[{"\[ExponentialE]", "-", \(\(\[ExponentialE]\ x\)\/2\), "+", InterpretationBox[\(O[x]\^2\), SeriesData[ x, 0, {}, 0, 2, 1]]}], SeriesData[ x, 0, {E, Times[ Rational[ -1, 2], E]}, 0, 2, 1]]], "Output"] }, Open 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