I see the problem $\limsup_{n\to\infty}({sin(n)}), \liminf_{n\to\infty}({sin(n)})$. And i prove it with dirichlet approximation. Also Through this, i prove $\liminf_{n\to\infty}({n\ sin^2(n)})=0$. so i think what is $\limsup_{n\to\infty}({n^2 \sin^2(n)})$. but i cannnot do it. How can i know $\limsup_{n\to\infty}({n^2 \sin^2(n)})$. is there other theorems like dirichlet approximation theorem?
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https://mathoverflow.net/questions/83550/has-n2sinn-limit Related. Also see https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown/20609#20609. – Ningxin May 20 '20 at 09:06
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It's probably $0$, but if the irrationality measure of $\pi$ is exactly $2$ it might be positive. – Daniel Fischer May 20 '20 at 14:33