Re: Question on Sum[] function

• To: mathgroup at smc.vnet.net
• Subject: [mg86631] Re: Question on Sum[] function
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Sat, 15 Mar 2008 17:43:10 -0500 (EST)
• References: <fr5du5\$o43\$1@smc.vnet.net> <fras4i\$18m\$1@smc.vnet.net>

```On Thu, 13 Mar 2008, David W.Cantrell wrote:
> sigmundv at gmail.com wrote:
>> Dear group members,
>>
>> If I evaluate Sum[Log[x^n],{n,1,Infinity}] I get -Log[x]/12 as an
>> answer, but if I plug any other value than -1 or +1 in, Mathematica
>> tells me that the series is divergent; for x=-1 the sum is -I*Pi/12
>> and for x=1 the sum is 0, of course.
>>
>> If we restrict ourselves to real numbers, I would say that the series
>> is only meaningful for x>0, because for x<0, Log[x^n] is not defined
>> for odd n. For x>0, we write Sum[Log[x^n],{n,1,Infinity}] as
>> Sum[n*Log[x],{n,1,Infinity}], and clearly this series is only
>> convergent for x=1, with sum 0.
>>
>> Well, my actual question was how to interpret the closed form
>> expression that Mathematica gives for the sum of the afore-mentioned
>> series. Mathematica ought to return to me some condition on x, because
>> Sum[Log[x^n],{n,1,Infinity}] == -Log[x]/12 is not true for all real,
>> or complex, x.
>>
>> I hope that you can shed some light on this.
>
> Since nobody has answered yet...
>
> For simplicity, let's assume that x > 0, so that, as you said, we may
> "write Sum[Log[x^n],{n,1,Infinity}] as Sum[n*Log[x],{n,1,Infinity}]".
> If x is not 1, then the latter sum is formally
>
> Log[x] * Sum[n,{n,1,Infinity}]                            (*)
>
> Now of course Sum[n,{n,1,Infinity}] diverges to Infinity. But there is a
> sense in which it equals -1/12:
>
> Consider
>
> In[6]:= Sum[n^p, {n, 1, Infinity}]
>
> Out[6]= Zeta[-p]
>
> Now of course if p is 1, the sum in In[6] is Sum[n,{n,1,Infinity}].
> Replacing p by 1 in Out[6], we get
>
> In[7]:= Zeta[-1]
>
> Out[7]= -1/12
>
> "Therefore", Sum[n,{n,1,Infinity}] equals -1/12. Substituting that result
> into (*), we get
>
> -Log[x]/12
>
> which is the result which Mathematica had given for your original sum.
>
> You're welcome to think of what I said as being rather absurd. Yet, I
> suspect that what I said is actually related to the reason that
> Mathematica gave -Log[x]/12 for your sum.
>
> David
>
Hello Sigmund and David,

The behavior of Sum in the above example is unintentional. As noted
by David, the factor of (-1/12) in the answer comes from Zeta[-1].

The expected behavior for this sum would be for it to return unevaluated,
except when x = 1.