
Re: bounded sequence or not ?
Posted:
Jun 27, 2014 8:08 AM


On Thursday, June 26, 2014 9:50:26 AM UTC4, 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 wellknown 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).

