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,
>
> -----
> 	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

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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