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?
• 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,
>
> -----
> 	Department of Mathematics and Statistics
> 	Southern Illinois University Edwardsville
> aweyhau at siue.edu
> http://www.siue.edu/~aweyhau/
> Science Building 1316	(618) 650-2220
>
>
>

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,

Wolfram Research, Inc.

```

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