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Re: convert an expression to an infinite series

  • To: mathgroup at
  • Subject: [mg29632] Re: [mg29618] convert an expression to an infinite series
  • From: Andrzej Kozlowski <andrzej at>
  • Date: Fri, 29 Jun 2001 01:36:05 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

on 01.6.28 6:28 PM, Toshiyuki (Toshi) Meshii at meshii at

> Hello,
> Does anyone know a way to convert an expression to an infinite series?
> Let me show the idea by a concrete problem.
> In[29]:=
> Sum[E^(-2*m*n*L), {n, 0, Infinity}]
> Out[29]=
> E^(2*L*m)/(-1 + E^(2*L*m))
> Yes, Mathematica 4.1 knows that the infinite sum converges to a specific
> value.
> Then, my question is that how can I expand the expression
> E^(2*L*m)/(-1 + E^(2*L*m))
> in an infinite series on n, in this case
> Sum[E^(-2*m*n*L), {n, 0, Infinity}]
> Is there any way or is it an one way path?
> -Toshi

First of all, if Mathematica returned Sum[E^(-2*m*n*L), {n, 0, Infinity}]
than it would of course immediately evaluate it back to E^(2*L*m)/(-1 +
E^(2*L*m)) and you would be back where you started. That of course could be
dealt with by using HoldForm (for example). Secondly, it would be extremely
difficult  (perhaps impossible) to implement what you want in sufficient
generality for it to be useful. (I take that you would like Mathematica to
return a power-series in terms of a  general  "n-th term". One can certainly
implement a limited version of this and there may be a package that does it
but  the following method is actually more useful. (Copy the  expression
below the signature and paste into a Mathematica notebook).

Andrzej Kozlowski
Toyama International University

(************** 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
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one of the following:

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  with a name ending in .nb, then open the file inside the

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  clipboard, then use the Paste menu command inside the application.

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(*CacheID: 232*)

(*NotebookOptionsPosition[      2817,         87]*)
(*NotebookOutlinePosition[      3652,        113]*)
(*  CellTagsIndexPosition[      3608,        109]*)


    \(v = \[Sum]\+\(n = 0\)\%\[Infinity] \[ExponentialE]\^\(\(-2\)\ m\
\ n\ L\)\)], "Input"],

    \(TraditionalForm\`\[ExponentialE]\^\(2\ L\ m\)\/\(\(-1\) + \
\[ExponentialE]\^\(2\ L\ m\)\)\)], "Output"]
}, Open  ]],


    \(\((v /. \[InvisibleSpace]\[ExponentialE]\^\(2\ L\ m\) \[Rule]
              x)\) + O[x]\^5 /. \[InvisibleSpace]x \[Rule] \
\[ExponentialE]\^\(2\ L\ m\)\)], "Input"],

        RowBox[{\(-\[ExponentialE]\^\(2\ L\ m\)\),
          "-", \(\((\[ExponentialE]\^\(2\ L\ m\))\)\^2\),
          "-", \(\((\[ExponentialE]\^\(2\ L\ m\))\)\^3\),
          "-", \(\((\[ExponentialE]\^\(2\ L\ m\))\)\^4\), "+",
          InterpretationBox[\(O(\((\[ExponentialE]\^\(2\ L\ \
              Power[ E,
                Times[ 2, L, m]], 0, {}, 1, 5, 1]]}],
          Power[ E,
            Times[ 2, L, m]], 0, {-1, -1, -1, -1}, 1, 5, 1]],
      TraditionalForm]], "Output"]
}, Open  ]]
FrontEndVersion->"4.1 for Macintosh",
ScreenRectangle->{{0, 832}, {0, 604}},
WindowSize->{520, 482},
WindowMargins->{{Automatic, 25}, {-10, Automatic}},

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.  The cache data will then be recreated when
you save this file from within Mathematica.




Cell[1727, 52, 106, 2, 49, "Input"],
Cell[1836, 56, 124, 2, 45, "Output"]
}, Open  ]],

Cell[1997, 63, 185, 3, 31, "Input"],
Cell[2185, 68, 616, 16, 31, "Output"]
}, Open  ]]

End of Mathematica Notebook file.

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