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[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 Derivative[4][Zeta][2]. 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 know. David W. Cantrell

**Follow-Ups**:**Re: Re: Why does Sum return 0 on this series? Bug Report***From:*Syd Geraghty <sydgeraghty@mac.com>