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