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MathGroup Archive 2008

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Question on Sum[] function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg86417] Question on Sum[] function
  • From: sigmundv at gmail.com
  • Date: Tue, 11 Mar 2008 02:55:58 -0500 (EST)

Dear group members,

If I evaluate Sum[Log[x^n],{n,1,Infinity}] I get -Log[x]/12 as an
answer, but if I plug any other value than -1 or +1 in, Mathematica
tells me that the series is divergent; for x=-1 the sum is -I*Pi/12
and for x=1 the sum is 0, of course.

If we restrict ourselves to real numbers, I would say that the series
is only meaningful for x>0, because for x<0, Log[x^n] is not defined
for odd n. For x>0, we write Sum[Log[x^n],{n,1,Infinity}] as
Sum[n*Log[x],{n,1,Infinity}], and clearly this series is only
convergent for x=1, with sum 0.

Well, my actual question was how to interpret the closed form
expression that Mathematica gives for the sum of the afore-mentioned
series. Mathematica ought to return to me some condition on x, because
Sum[Log[x^n],{n,1,Infinity}] == -Log[x]/12 is not true for all real,
or complex, x.

I hope that you can shed some light on this.

Kind regards,
Sigmund Vestergaard


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