I need to find the $\limsup$ and $\liminf$ of the following sequence $$ x_n = \sin\left(\frac{n\pi}{2}\right) \cos\left(\frac{n\pi}{3}\right). $$ I was trying to take subsequences as $n_k=12k, 12k+1, 12k+2,.......,12k+11$ but this will take long time to evaluate them. Is there a faster way to find the values of $\limsup$ and $\liminf$?
2026-04-09 04:27:13.1775708833
Find limit sup and limit inf for the sequence $ x_n = \sin(\frac{n\pi}{2}) \cos(\frac{n\pi}{3})$
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We can say that: $$\sin\left(\frac{n\pi}{2}\right)=\left\{0,1,0,-1,0,1,0\right\} \,\,\, n=0,\dots,6$$ And: $$\cos\left(\frac{n\pi}{3}\right)=\left\{1,\frac{1}{2},-\frac{1}{2},-1,-\frac{1}{2},\frac{1}{2},1\right\}\,\,\, n=0,\dots,6$$
So: $$\sup(x_n)=\max(x_n)=1$$ And: $$\inf(x_n)=\min(x_n)=0$$