Re: Wrong Limit

• To: mathgroup at smc.vnet.net
• Subject: [mg47790] Re: [mg47775] Wrong Limit
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 27 Apr 2004 04:46:33 -0400 (EDT)
• References: <200404260641.CAA06357@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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.

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.
> Thanks.
>
>
>

```

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