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Re: Why does Sum return 0 on this series?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82618] Re: [mg82537] Why does Sum return 0 on this series?
  • From: Devendra Kapadia <dkapadia at wolfram.com>
  • Date: Fri, 26 Oct 2007 05:23:26 -0400 (EDT)
  • References: <200710240824.EAA05440@smc.vnet.net>

On Wed, 24 Oct 2007, 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
>
>
>
Hello Adam,

Thank you for reporting the problem with the incorrect answer
for the above infinite sum.

The basic idea for evaluating this logarithmic sum is to
replace the Log term with a power of n since we have:

====================

In[1]:= $Version

Out[1]= 6.0 for Linux x86 (32-bit) (June 28, 2007)

In[2]:= D[n^x, {x, 4}]/.{x-> 0}//InputForm

Out[2]//InputForm= Log[n]^4

===================

The incorrect answer 0 arises while using the result from a
series expansion internally, after applying the above
replacement in the sum.

The correct answer, Derivative[4][Zeta][2], may be obtained
as follows:

==========================

In[3]:= Sum[n^x/n^2, {n,2, Infinity}]

Out[3]= -1 + Zeta[2 - x]

In[4]:= D[%, {x,4}]/.{x-> 0}//InputForm

Out[4]//InputForm= Derivative[4][Zeta][2]

In[5]:= N[%, 11]

Out[5]= 24.001486394

In[6]:= NSum[Log[n]^4/n^2, {n, 2, Infinity}, WorkingPrecision->20]

Out[6]= 24.001486394

=========================

I apologize for the inconvenience caused by this problem.

Sincerely,

Devendra Kapadia,
Wolfram Research, Inc.


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