Stuck at proving whether the sequence is convergent or not

It happens that your sequence is a cyclic sequence. More precisely, its first six terms are$$-2+\frac{\sqrt3}2,2+\frac{\sqrt3}2,4,2-\frac{\sqrt3}2,-2-\frac{\sqrt3}2,\text{ and }-4$$and then it repeats itself again and again. Therefore:

• $\max\{X_n\,|\,n\in\mathbb N\}=\sup\{X_n\,|\,n\in\mathbb N\}=\limsup_nX_n=4$;
• $\min\{X_n\,|\,n\in\mathbb N\}=\inf\{X_n\,|\,n\in\mathbb N\}=\liminf_nX_n=-4$;
• the sequence diverges.

The limit of a sequence, if it exists, is equal to its lim inf and lim sup. Accordingly, if the lim inf and lim sup of a sequence are different, then its limit cannot exist.

Computing the first $6$ terms alone does not prove that $\min X_n=-4$ nor $\max X_n=4$ but rather that $\min X_n\le-4$ and $\max X_n \ge 4$ as it does not say anything about the rest of the sequence.

Hint : $X_{n+6}=X_n$