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MathGroup Archive 1998

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Re: Series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14181] Re: Series
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 30 Sep 1998 19:42:16 -0400
  • Organization: University of Western Australia
  • References: <6ushsl$5on@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

MAvalosJr at aol.com wrote:

> I know that merely giving a finite number of terms of a sequence or
> series does not define a unique nth term. In fact, an infinite number
> of nth terms are possible. However, any suggestions on how I could
> write an nth term for a particular sequence using mathematica? For
> example: 1/1! - 1/2! + 1/3! -1/4!+......

A neat way of finding patterns in numbers is to use Sloane's On-Line
Encyclopedia of Integer Sequences at

	http://www.research.att.com/~njas/sequences/

See also the Notebook appended below and the Mathematica Journal 5(3):
33-35.

Cheers,
	Paul 
 
____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at physics.uwa.edu.au  AUSTRALIA                       
http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________

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general this is impossible as there are an infinite number of functions
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\
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Let us apply this method to the truncated series (polynomial)\ \>", 
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Cell[BoxData[
    \(TraditionalForm\`\((\(-x\))\)\^n\/\(n!\)\)], "Output"] }, Open 
]],

Cell["The (infinite) series reduces to:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"Factor", "[", 
        RowBox[{\(\[Sum]\+\(n = 0\)\%\[Infinity]\), 
          FormBox["%",
            "TraditionalForm"]}], "]"}], TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\[ExponentialE]\^\(-x\)\)], "Output"] }, Open 
]],

Cell["\<\
Alternatively, we can compute the sum directly as a hypergeometric \
function:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      TagBox[
        RowBox[{\(\(\[ThinSpace]\_0\) F\_0\), "(", 
          RowBox[{
            TagBox[
              InterpretationBox["\[InvisibleSpace]",
                {}],
              (Editable -> False)], ";", 
            TagBox[
              InterpretationBox["\[InvisibleSpace]",
                {}],
              (Editable -> False)], ";", 
            TagBox[\(-x\),
              (Editable -> True)]}], ")"}],
        InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&]], 
      TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\[ExponentialE]\^\(-x\)\)], "Output"] }, Open 
]],

Cell["As a check, we can expand this into a Maclaurin series:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"%", "-", " ", "\[ScriptS]", " ", "+", " ", 
        FormBox[\(\(O(x)\)\^6\),
          "TraditionalForm"]}], TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      InterpretationBox[\(O(x\^6)\),
        SeriesData[ x, 0, {}, 6, 6, 1]], TraditionalForm]], "Output"] },
Open  ]],

Cell["Note that this method is not restricted to numerical series:",
"Text"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\[ScriptS] = 
      1 - \(\[Alpha]\ x\)\/\(\[Alpha] + 1\) + 
        \(\[Alpha]\ x\^2\)\/\(\(2!\)\ \((\[Alpha] + 2)\)\) - 
        \(\[Alpha]\ x\^3\)\/\(\(3!\)\ \((\[Alpha] + 3)\)\); \)\)],
"Input"],

Cell["The successive ratios are simple:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`\[ScriptR] = ratios(%, x)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-\(\[Alpha]\/\(\[Alpha] + 1\)\)\), 
      \(-\(\(\[Alpha] + 1\)\/\(\[Alpha] + 2\)\)\), 
      \(-\(\(\[Alpha] + 2\)\/\(\[Alpha] + 3\)\)\)}\)], "Output"] }, Open
]],

Cell["and with", "Text"],

Cell[BoxData[
    \(TraditionalForm\`\(p = \(q = 1\); \)\)], "Input"],

Cell["the matrix reads", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"BlockMatrix", "(", 
        RowBox[{"(", GridBox[{
              {"\[ScriptCapitalP]", "\[ScriptCapitalQ]"}
              }], ")"}], ")"}], TraditionalForm]], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {"1", "1", \(\[Alpha]\/\(\[Alpha] + 1\)\)},
            {"1", "2", \(\(2\ \((\[Alpha] + 1)\)\)\/\(\[Alpha] + 2\)\)},
            {"1", "3", \(\(3\ \((\[Alpha] + 2)\)\)\/\(\[Alpha] + 3\)\)}
            }], ")"}], TraditionalForm]], "Output"] }, Open  ]],

Cell["Solving the linear system of equations:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`LinearSolve(%, \[ScriptR]) // Factor\)],
"Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-\(\(\[Alpha] - 1\)\/\[Alpha]\)\), \(-\(1\/\[Alpha]\)\),
1\/\[Alpha]}
      \)], "Output"]
}, Open  ]],

Cell["the rational function is", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\(Take(%, p + 1)\) . 
          \(Table(n\^i, {i, 0, p})\)\/\(\(Take(%, \(-q\))\) . 
            \(Table(n\^i, {i, q})\) + 1\) // Factor\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\(n + \[Alpha] - 1\)\/\(n +
\[Alpha]\)\)\)\)], 
  "Output"]
}, Open  ]],

Cell["and the general term of the series reads:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\[ScriptC] = 
      \(% /. b_.  + n\ a_Integer \[Rule] a\) /. n + b_.  \[Rule] 1\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(-1\)\)], "Output"] }, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"Simplify", "(", 
        FractionBox[
          RowBox[{
            RowBox[{"(", 
              RowBox[{\(%%\/\[ScriptC]\), "/.", 
                RowBox[{\(n\ a_.  + b_. \), "\[Rule]", 
                  RowBox[{"a", " ", 
                    TagBox[\(\((b\/a + 1)\)\_n\),
                      Pochhammer]}]}]}], ")"}], " ", 
            \(\((\[ScriptC]\ x)\)\^n\)}], \(n!\)], ")"}],
TraditionalForm]], 
  "Input"],

Cell[BoxData[
    FormBox[
      FractionBox[
        RowBox[{\(\((\(-x\))\)\^n\), " ", 
          TagBox[\(\((\[Alpha])\)\_n\),
            Pochhammer]}], 
        RowBox[{\(n!\), " ", 
          TagBox[\(\((\[Alpha] + 1)\)\_n\),
            Pochhammer]}]], TraditionalForm]], "Output"] }, Open  ]],

Cell["which leads to the hypergeometric function", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      TagBox[
        RowBox[{\(\(\[ThinSpace]\_1\) F\_1\), "(", 
          RowBox[{
            TagBox[
              TagBox[
                TagBox["\[Alpha]",
                  (Editable -> True)],
                InterpretTemplate[ {
                  SlotSequence[ 1]}&]],
              (Editable -> False)], ";", 
            TagBox[
              TagBox[
                TagBox[\(\[Alpha] + 1\),
                  (Editable -> True)],
                InterpretTemplate[ {
                  SlotSequence[ 1]}&]],
              (Editable -> False)], ";", 
            TagBox[\(-x\),
              (Editable -> True)]}], ")"}],
        InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&]], 
      TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`x\^\(-\[Alpha]\)\ 
      \((\[CapitalGamma](\[Alpha] + 1) - 
          \[Alpha]\ \(\[CapitalGamma](\[Alpha], x)\))\)\)], "Output"] },
Open  ]],

Cell["Alternatively, the infinite series yields", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{\(\[Sum]\+\(n = 0\)\%\[Infinity]\), 
        FormBox[
          FractionBox[
            RowBox[{\(\((\(-x\))\)\^n\), " ", 
              TagBox[\(\((\[Alpha])\)\_n\),
                Pochhammer]}], 
            RowBox[{\(n!\), " ", 
              TagBox[\(\((\[Alpha] + 1)\)\_n\),
                Pochhammer]}]],
          "TraditionalForm"]}], TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\[Alpha]\ 
      \((x\^\(-\[Alpha]\)\ \(\[CapitalGamma](\[Alpha])\) - 
          x\^\(-\[Alpha]\)\ \(\[CapitalGamma](\[Alpha], x)\))\)\)],
"Output"]
}, Open  ]],

Cell["\<\
With some work this could be turned into a package for converting \ the
first few terms of a (hypergeometric) series into the corresponding \
(minimal) hypergeometric function.\
\>", "Text"]
}, Open  ]]
}]


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