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Cnop Vrije Universiteit Brussel icnop@vub.ac.be\ \>", "Subsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell["How fast does the Stirling formula approach the factorial?", \ "Subtitle"], Cell[CellGroupData[{ Cell["The Stirling formula", "Section"], Cell["Here is the Stirling approximation for n!", "Text"], Cell[BoxData[ \(TraditionalForm\`s( n_) := \@\(2\ \[Pi]\ n\)\ \((n\/\[ExponentialE])\)\^n\)], "Input"], Cell["Check a few values:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`s(4. )\)], "Input"], Cell[BoxData[ \(23.506175132893294`\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(4!\)\)], "Input"], Cell[BoxData[ \(24\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[s[1000. ]]]\)], "Input"], Cell[BoxData[ \(0.009999999999999787`\ Second\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[\(1000!\)]]\)], "Input"], Cell[BoxData[ \(0.`\ Second\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`\(s(1000. )\)\/\(1000!\)\)], "Input"], Cell[BoxData[ \(0.999916670141105118072439`12.0236\)], "Output"] }, Closed]], Cell["which is close to 1 , while both numbers are huge:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`N[s(1000)]\)], "Input"], Cell[BoxData[ \(4.02353729203698903811248`12.9537*^2567\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(\(1000!\)\)], "Input"], Cell[BoxData[ \(402387260077093773543702433923003985719374864210714632543799910429938512\ 398629020592044208486969404800479988610197196058631666872994808558901323829669\ 944590997424504087073759918823627727188732519779505950995276120874975462497043\ 601418278094646496291056393887437886487337119181045825783647849977012476632889\ 835955735432513185323958463075557409114262417474349347553428646576611667797396\ 668820291207379143853719588249808126867838374559731746136085379534524221586593\ 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It takes some time to generate:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(First[ Timing[Table[ 100\ N[\((\(-s[n]\) + \(n!\))\)/\(n!\)], {n, 20, 8000, 10}]]]\)], "Input"], Cell[BoxData[ \(47.9666666666667`\ Second\)], "Output"] }, Closed]], Cell["\<\ Make plot with larger step size to cover more of the x -axis \ :\ \>", "Text"], Cell[BoxData[ \(\(relative = Table[{n, 100\ N[\((\(-s[n]\) + \(n!\))\)/\(n!\)]}, {n, 2000, 50000, 1000}];\)\)], "Input"], Cell["and try to approximate with rational expressions", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Fit[relative, {1, 1/x, 1/x^2, 1/x^3}, x]\)], "Input"], Cell[BoxData[ \(\(-2.909881883168165`*^-10\) + 18.20591836259098`\/x\^3 - 0.3631212810198137`\/x\^2 + 8.33333724649616`\/x\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Fit[relative, {1, 1/x, 1/x^2}, x]\)], "Input"], Cell[BoxData[ \(\(-2.3366658261093594`*^-10\) - 0.35044257495907283`\/x\^2 + 8.333335229017969`\/x\)], "Output"] }, Closed]], Cell["\<\ Only the 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