       Re: Why does Sum return 0 on this series?

• To: mathgroup at smc.vnet.net
• Subject: [mg82586] Re: Why does Sum return 0 on this series?
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Thu, 25 Oct 2007 06:11:42 -0400 (EDT)
• References: <ffmvp0\$5er\$1@smc.vnet.net>

```Adam Weyhaupt <aweyhau at siue.edu> wrote:
> In Mathematica 6.0.1 on Mac OS X, Sum returns 0 on this series.
>
> In:= Sum[Log[n]^4/n^2, {n, 2, Infinity}]
> Out= 0
> In:= \$Version
> Out= "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[Zeta].

In:= N[Derivative[Zeta], 11]

Out= 24.001486394

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

Out= 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:= Sum[Log[n]^4/n^2, {n, 3, Infinity}]

Out= -(1/4)*Log^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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