Let $(a\_n)\_n$ be an increasing real sequence with $a\_n=O(\sqrt n)$.


Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits\_{k=1}^{\phi(n)}\cos(a\_k)\right|+\left|\sum\limits\_{k=1}^{\phi(n)}\sin(a\_k)\right|=\infty?$$


<https://artofproblemsolving.com/community/c7h3080371_majoration_of_cosinus_and_sinus_addition>



Series involving sine and cosine
Yes, it suffices that $a\_n=o(n)$.


Denote $h(n)=|\sum\_{j=1}^n \cos a\_j|+|\sum\_{j=1}^n \sin a\_j|$.


For any fixed integer $M>0$ there exists arbitrarily large $n$ for which $a\_{n+M}-a\_n<1/M^2$. Thus by triangle inequality and 1-Lipschitz property of functions $\cos$ and $\sin$ we have
$$h(n+M)+h(n)\geqslant |\cos a\_{n+1}+\ldots+\cos a\_{n+M}|+|\sin a\_{n+1}+\ldots+\sin a\_{n+M}|\\\geqslant M|\cos a\_{n+1}|-M/M^2+M|\sin a\_{n+1}|-M/M^2\geqslant 2M-2$$
that yields that $h(n+M)$ or $h(n)$ is at least $M-1$. Your claim follows immediately.


