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Re: Why does Sum return 0 on this series?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg82585] Re: [mg82537] Why does Sum return 0 on this series?
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 25 Oct 2007 06:11:11 -0400 (EDT)
*References*: <200710240824.EAA05440@smc.vnet.net>
On 24 Oct 2007, at 17:24, Adam Weyhaupt wrote:
> In Mathematica 6.0.1 on Mac OS X, Sum returns 0 on this series.
>
> In[1]:= Sum[Log[n]^4/n^2, {n, 2, Infinity}]
> Out[1]= 0
> In[2]:= $Version
> Out[2]= "6.0 for Mac OS X x86 (32-bit) (June 19, 2007)"
>
> Of course that's not right, because the terms are all positive. I
> don't expect Sum to always return a correct answer for any series
> (that would be unreasonable), and NSum is the more appropriate
> command to investigate this series numerically. But 0 seems like a
> very strange answer. Can anyone provide any insight on why
> Mathematica is returning 0 for the sum of a positive series?
>
> Thanks,
>
> Adam
> -----
> Adam Weyhaupt
> Department of Mathematics and Statistics
> Southern Illinois University Edwardsville
> aweyhau at siue.edu
> http://www.siue.edu/~aweyhau/
> Science Building 1316 (618) 650-2220
>
>
>
It's a bug, of course. Bugs happen, even in Mathematica. In fact the
answer ought to be:
Derivative[4][Zeta][2]
Derivative[4][Zeta][2]
N[%, 10]
24.001486393736461571`9.999999999999998
This agrees with the answer given by NSum:
NSum[Log[n]^4/(n)^2, {n, 2, Infinity}, PrecisionGoal -> 10,
WorkingPrecision -> 20]
24.001486394
You can prove this using the first series here:
http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/
Zeta2/06/05/01/02/
Andrzej Kozlowski
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