I feel in this case I'm seeing with every tail that the limsup and liminf are both infinity but it not certain. Any guidance please !
2026-03-29 05:58:44.1774763924
Determine limsup and liminf of the seqeuence $a_n = n\cos \left( \frac{\pi n}{4} \right)$
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$a_{8n} = 8n, a_{8n+4} = -8n$. It follows that $\limsup a_n = \infty, \liminf_n a_n = -\infty$.