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Re: Re: Why does Sum return 0 on this series? Bug Report

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82613] Re: [mg82586] Re: Why does Sum return 0 on this series? Bug Report
  • From: Syd Geraghty <sydgeraghty at mac.com>
  • Date: Fri, 26 Oct 2007 05:20:52 -0400 (EDT)
  • References: <ffmvp0$5er$1@smc.vnet.net> <200710251011.GAA21911@smc.vnet.net>

David,

The bug persists in Mathematica V6.0.1

In[1]=Sum[Log[n]^4/n^2, {n, 3, Infinity}]

Out[1]= -(1/4) Log[2]^4


Cheers ... Syd

Syd Geraghty B.Sc., M.Sc.
sydgeraghty at mac.com
San Jose, CA

My System

Mathematic 6.0.1 for Mac OS X x86 (32 - bit) (June 19, 2007)
MacOS X V 10.4 .10
MacBook Pro 2.33 Ghz Intel Core 2 Duo  2GB RAM

On Oct 25, 2007, at 3:11 AM, David W.Cantrell wrote:

> Adam Weyhaupt <aweyhau at siue.edu> 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),
>
> OTOH, I think it would be reasonable that Sum should always return a
> correct answer for any series, assuming that we consider returning the
> original series unevaluated as being one type of correct answer.
>
>> 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?
>
> Sorry, I can't do that. However, you might be interested to know  
> that that
> sum can be expressed in closed form in Mathematica; it's
>
> Derivative[4][Zeta][2].
>
> In[2]:= N[Derivative[4][Zeta][2], 11]
>
> Out[2]= 24.001486394
>
> In[3]:= NSum[Log[n]^4/n^2, {n, 2, Infinity}, WorkingPrecision->20]
>
> Out[3]= 24.001486394
>
> --------------------------
>
> BTW, at least in version 5.2, we can get negative values for some  
> sums of
> positive terms, related to your original series. For example,
>
> In[14]:= Sum[Log[n]^4/n^2, {n, 3, Infinity}]
>
> Out[14]= -(1/4)*Log[2]^4
>
> Whether the source of that bug is the same as the source of yours,  
> I don't
> know.
>
> David W. Cantrell
>



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