On Thursday, June 26, 2014 9:50:26 AM UTC-4, Port563 wrote: > "troisquatorze" <email@example.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).