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Topic: bounded sequence or not ?
Replies: 30   Last Post: Jul 1, 2014 2:53 AM

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 Pubkeybreaker Posts: 1,683 Registered: 2/12/07
Re: bounded sequence or not ?
Posted: Jun 27, 2014 8:08 AM

On Thursday, June 26, 2014 9:50:26 AM UTC-4, Port563 wrote:
> "troisquatorze" <bbaborum@gmail.com> wrote... > > One puts S_n=sum(cos(k^2),k=0..n) > Is (S_n) bounded or not ?
>>Well if n is finite, of course it is bounded! Each term is finite (belonging to (-1,1]) and if there's a finite number of them.... etc. .... (-:
>>So, you meant the infinite sum (and "series", not "sequence"). There are many superficially similar problems (e.g. with cos(k^-2)) with well-known solutions, but this one is materially different and for the obvious reason
>> (think of the limit of the sequence (cf. series) term a_k (S_k = sum(a_0...a_k)) as k-->oo)
>> I tried applying Dirichlet's test and partial sums proofs but got nowhere

This is closely tied to the question of whether k^2 mod pi is uniformly
distributed. I seem to recall that there is a theorem about polynomial
ranges modulo irrationals but I just don't remember when/where I saw it.
It might have been during a lecture from back in the late 80's from Uncle Paul
(Erdos).

Date Subject Author
6/26/14 troisquatorze
6/26/14 Port563
6/26/14 Peter Percival
6/26/14 Jussi Piitulainen
6/26/14 Port563
6/26/14 Jussi Piitulainen
6/26/14 troisquatorze
6/26/14 troisquatorze
6/27/14 Pubkeybreaker
6/27/14 troisquatorze
6/26/14 troisquatorze
6/26/14 Roland Franzius
6/26/14 troisquatorze
6/26/14 quasi
6/27/14 Math Lover
6/27/14 quasi
6/27/14 Timothy Murphy
6/27/14 Port563
6/27/14 Math Lover
6/27/14 Timothy Murphy
6/27/14 Math Lover
6/27/14 quasi
6/27/14 Roland Franzius
6/27/14 Port563
6/27/14 quasi
6/29/14 quasi
6/29/14 quasi
6/30/14 troisquatorze
6/30/14 quasi
6/30/14 Roland Franzius
7/1/14 troisquatorze