Re: Wrong Limit

• To: mathgroup at smc.vnet.net
• Subject: [mg47843] Re: Wrong Limit
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Wed, 28 Apr 2004 06:56:41 -0400 (EDT)
• References: <200404260641.CAA06357@smc.vnet.net> <c6l76k\$ikg\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> First of all, Mathematica 5.0 gives:
>
> Sum[1/k^p, {k, 1, Infinity}]
>
> Zeta[p]
>
> which is correct so there is no real need to take any limits. Still,
> your observation seems to reveal a bug in Mathematica 5.0. which may be
> related to one that was already discussed here recently.

I see at the end of your post that you demonstrate a bug in getting
Limit[PolyGamma... If that was the bug discussed here recently, I seem to
have overlooked that thread. But in any event, Ray's observation also
reveals a bug in getting Limit[HarmonicNumber... Note first that we have,
correctly,

In[1]:= HarmonicNumber[Infinity, 3]

Out[1]= Zeta[3]

but then, incorrectly,

In[2]:= Limit[HarmonicNumber[n, 3], n -> Infinity]

Out[2]= Infinity

Perhaps the Limit[PolyGamma... bug and the Limit[HarmonicNumber... bug are
related. I don't know.

David Cantrell

> First, look at this:
>
> Sum[1/k^3,{k,1,m}]
>
> HarmonicNumber[m, 3] + PolyGamma[2, 1]/2 + Zeta[3]
>
> This looks strange, but in fact is correct since:
>
> FullSimplify[PolyGamma[2,1]/2+Zeta[3]]
>
> 0
>
> Now
>
> FullSimplify[Sum[1/k^3,{k,1,m}]]
>
> PolyGamma[2, 1 + m]/2 + Zeta[3]
>
> This is still correct:
>
> PolyGamma[2,1+m]/2==HarmonicNumber[m,3]+PolyGamma[2,1]/2//FullSimplify
>
> True
>
> However, it tells us that Limit[PolyGamma[2, 1 + m]/2,m->Infinity]
> ought to be zero, while Mathematica gives:
>
> Limit[PolyGamma[2,1+m]/2,m->Infinity]
>
> Infinity
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>
> On 26 Apr 2004, at 15:41, Ray wrote:
>
> > In Mathematica 4.2, if s[n_]:= Sum[1/k^p,{k,1,n}], then the output for
> > Limit[s[n],n->Infinity] was Limit[HarmonicNumber[n,p],n->Infinity].
> > Under 5.0.1, the answer to Limit[s[n],n->Infinity] is given incorrectly
> > as Infinity for odd p and the actual numerical value for even p. Anyone
> > know why Mathematica now gives an incorrect result here for odd p.

```

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