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
(***********************************************************************
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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"]
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Cell[BoxData[
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Cell[BoxData[
\(18872\)], "Output"]
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Cell["All these manipulations can be done by hand.", "Text"]
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