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, > > Adam > ----- > Adam Weyhaupt > 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 answer ought to be: 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

**References**:**Why does Sum return 0 on this series?***From:*Adam Weyhaupt <aweyhau@siue.edu>