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Re: Why does Sum return 0 on this series?

Adam Weyhaupt <aweyhau at> 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


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

David W. Cantrell

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