slightly OT: harmonic series revisited...

[franktaw at netscape.net]

That is an outline of a valid proof of divergence, but it really has nothing to do with the posted incorrect proof.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: Richard Guy <rkg at cpsc.ucalgary.ca>
To: Rob Arthan <rda at lemma-one.com>
Cc: santi_spadaro at virgilio.it; franktaw at netscape.net; hv at crypt.org; seqfan at ext.jussieu.fr
Sent: Tue, 13 Dec 2005 13:52:07 -0700 (MST)
Subject: Re: slightly OT: harmonic series revisited...


Have come into this rather late, so apologize 
if this is old hat. The nth partial sum is 
c1 \ln n - c2 \ln\ln n, regardless of 
ordering, which can only affect c1 and c2, 
for which trivial bounds suffice. 
 
R. 
 
On Tue, 13 Dec 2005, Rob Arthan wrote: 
 
> 
> On 13 Dec 2005, at 17:46, Rob Arthan wrote: 
> 
>> On Tuesday 13 Dec 2005 2:45 pm, franktaw at netscape.net >> wrote: 
>>> Actually, that argument doesn't quite work. The >>> convergence of the 
>>> original sequence is conditional, not absolute, so you >>> can't arbitrarily 
>>> reorder the terms and draw any conclusions about >>> convergence. 
>> >> But the reordering is harmless here. Hugo's argument >> gives diverging lower 
>> bound for the sum of the a_n from 1 to 6p (since the >> reordered subsequence 
>> contains all the negative terms). 
> 
> And I was having a mental aberration at the time I wrote > it. The reordering is not obviously harmless and it isn't > obvious (at least to me) how to get a rigorous argument > out of Hugo's estimate. 
> 
> Apologies, 
> 
> Rob. 
>> >>> -----Original Message----- 
>>> From: Rob Arthan <rda at lemma-one.com> 
>>> To: hv at crypt.org; santi_spadaro at virgilio.it >>> <santi_spadaro at virgilio.it> 
>>> Cc: seqfan at ext.jussieu.fr 
>>> Sent: Tue, 13 Dec 2005 13:19:12 +0000 
>>> Subject: Re: slightly OT: harmonic series revisited... 
>>> >>> On Tuesday 13 Dec 2005 12:26 pm, hv at crypt.org wrote: 
>>>> "santi_spadaro at virgilio.it" >>>> <santi_spadaro at virgilio.it> wrote: 
>>>> :Anybody knows an answer (and a neat way to show that >>>> the answer is 
>>>> :true)? 
>>>> : 
>>>> :"Define a_n = 1/n if n is composite and a_n = -(1/n) >>>> if n is 
>>>> :prime. Does the series of a_n (sum from n to infinity >>>> of a_n) diverges?" 
>>>> >>>> If P diverges, consider the set {p, 2p, 3p, 4p, 6p}; >>>> this avoids 
>>>> collisions for all odd primes p, and the contribution >>>> to A for these 5 
>>>> numbers is (-1 + 1/2 + 1/3 + 1/4 + 1/6)/p = 1/4p, so A >>>> > P/4, and so A 
>>>> again diverges. 
>>> >> >> >> > 
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