# About the sequence $\{\sin (n^2) \}_n$

A colleague of mine asked me if the behavior of the sequence $\{\sin (n^2) \}_n$ is known. In particular, does it converge? If not, what are its liminf and limsup?

Where can I find some hints or a discussion about these oscillating sequences?

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$\limsup a_n\neq \liminf a_n$, therfore $\lim{a_n}$ is not exist(Why we need Weyl's inequality ?) – Salech Alhasov Oct 17 '12 at 8:57

In virtue of the Weyl's inequality for quadratic polynomials and the Weyl's equidistribution Theorem, $e^{in^2}$ is equidistributed in the unit circle, so the sequence does not converge and limsup/liminf are $\pm 1$.