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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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    \(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\), "(", 
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            TagBox[
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        InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&]], 
      TraditionalForm]], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\[ExponentialE]\^\(-x\)\)], "Output"] }, Open 
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Cell["As a check, we can expand this into a Maclaurin series:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    FormBox[
      InterpretationBox[\(O(x\^6)\),
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Open  ]],

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

Cell[BoxData[
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    \`\(\[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[
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    \`{\(-\(\[Alpha]\/\(\[Alpha] + 1\)\)\), 
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Cell["and with", "Text"],

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

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

Cell[CellGroupData[{

Cell[BoxData[
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"Input"],

Cell[BoxData[
    \(TraditionalForm
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Cell["the rational function is", "Text"],

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

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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            \(\((\[ScriptC]\ x)\)\^n\)}], \(n!\)], ")"}],
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Cell[BoxData[
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Cell["which leads to the hypergeometric function", "Text"],

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Cell[BoxData[
    \(TraditionalForm
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      \((\[CapitalGamma](\[Alpha] + 1) - 
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Open  ]],

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

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      \((x\^\(-\[Alpha]\)\ \(\[CapitalGamma](\[Alpha])\) - 
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"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  ]]
}]