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MathGroup Archive 2001

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Re: Summation without Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27463] Re: Summation without Mathematica
  • From: Hendrik van Hees <h.vanhees at gsi.de>
  • Date: Tue, 27 Feb 2001 00:37:17 -0500 (EST)
  • Organization: GSI Darmstadt
  • References: <97a7hg$al7@smc.vnet.net> <97cet4$dad@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

It was to sum

In[1]:= Sum[x^3(7/8)^x,{x,1,Infinity}]

Out[1]= 18872

If I understand the question right, you want to know how to obtain the
result by hand. I'd do it in the following way given in the Mathematica
notebook in the attachment (I hope it's not forbidden to attach Mathematica 
notebooks in this newsgroup).


-- 
Hendrik van Hees		Phone:  ++49 6159 71-2751
c/o GSI-Darmstadt SB3 3.183	Fax:    ++49 6159 71-2990
Planckstr. 1			mailto:h.vanhees at gsi.de
D-64291 Darmstadt		http://theory.gsi.de/~vanhees/index.html

(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 4.0,
MathReader 4.0, or any compatible application. 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).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
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***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[      3084,        112]*)
(*NotebookOutlinePosition[      3743,        136]*)
(*  CellTagsIndexPosition[      3699,        132]*)
(*WindowFrame->Normal*)



Notebook[{
Cell["Define an auxiliary function", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(g[a_, x_] = \((7/8)\)^\((a\ x)\)\)], "Input"],

Cell[BoxData[
    \(\((7\/8)\)\^\(a\ x\)\)], "Output"]
}, Open  ]],

Cell["Its 3rd derivative is", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(D[g[a, x], {a, 3}]\)], "Input"],

Cell[BoxData[
    \(\(-\((7\/8)\)\^\(a\ x\)\)\ x\^3\ Log[8\/7]\^3\)], "Output"]
}, Open  ]],

Cell["\<\
The sum of g[a,x] over x (it's well known, because it's a geometric \
series):\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[a_] = Sum[\((7/8)\)^\((a\ x)\), {x, 1, Infinity}]\)], "Input"],

Cell[BoxData[
    \(\(-\(\((7\/8)\)\^a\/\(\(-1\) + \((7\/8)\)\^a\)\)\)\)], "Output"]
}, Open  ]],

Cell["\<\
You get the desired result by taking the 3rd derivative of the sum \
wrt. a (prove why, we interchanged sum and derivative at this point!) and \
taking a=1 at the end\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(h[a_] = FullSimplify[\(-D[f[a], {a, 3}]\)/Log[8/7]^3]\)], "Input"],

Cell[BoxData[
    \(\(7\^\(\(-4\)\ a\)\ \((4\^\(1 + 3\ a\)\ 49\^a + 2744\^a + \
3584\^a)\)\)\/\((\(-1\) + \((8\/7)\)\^a)\)\^4\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(h[1]\)], "Input"],

Cell[BoxData[
    \(18872\)], "Output"]
}, Open  ]],

Cell["All these manipulations can be done by hand.", "Text"]
},
FrontEndVersion->"4.0 for X",
ScreenRectangle->{{0, 1024}, {0, 768}},
WindowSize->{719, 600},
WindowMargins->{{Automatic, 49}, {58, Automatic}},
Magnification->1.5
]


(***********************************************************************
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.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1717, 49, 44, 0, 46, "Text"],

Cell[CellGroupData[{
Cell[1786, 53, 65, 1, 39, "Input"],
Cell[1854, 56, 54, 1, 60, "Output"]
}, Open  ]],
Cell[1923, 60, 37, 0, 46, "Text"],

Cell[CellGroupData[{
Cell[1985, 64, 51, 1, 39, "Input"],
Cell[2039, 67, 79, 1, 61, "Output"]
}, Open  ]],
Cell[2133, 71, 102, 3, 46, "Text"],

Cell[CellGroupData[{
Cell[2260, 78, 84, 1, 39, "Input"],
Cell[2347, 81, 84, 1, 86, "Output"]
}, Open  ]],
Cell[2446, 85, 190, 4, 71, "Text"],

Cell[CellGroupData[{
Cell[2661, 93, 86, 1, 39, "Input"],
Cell[2750, 96, 139, 2, 82, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2926, 103, 37, 1, 39, "Input"],
Cell[2966, 106, 39, 1, 39, "Output"]
}, Open  ]],
Cell[3020, 110, 60, 0, 46, "Text"]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)


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