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Re: Wrong Limit
*To*: mathgroup at smc.vnet.net
*Subject*: [mg47872] Re: Wrong Limit
*From*: Ray <rayfg at optonline.net>
*Date*: Thu, 29 Apr 2004 03:05:47 -0400 (EDT)
*References*: <200404260641.CAA06357@smc.vnet.net> <c6l76k$ikg$1@smc.vnet.net> <c6o4n9$cl8$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Possibly it has been noticed by others, but Mathematica 5.0.1 is unable
to compute HarmonicNumber[n]//N or HarmonicNumber[n,r]//N for large n,
say n>10^6 in any reasonable amount of time. However such problems could
be done instantaneously in Mathematica 4.2. Might be why the new version
is having trouble with the limit?
David W. Cantrell wrote:
> 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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