I can plug in huge numbers and it seems to tend to 4, but I don't know how I can prove this.
2026-03-28 15:25:58.1774711558
How can I prove the convergence/divergence of this?
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If you multiply everything out, the term of the highest power of $n$ in the numerator is $32n^5$. The highest power term in the bottom is $8n^5.$ So if you multiply everything out, and divide the top and bottom by $n^5$ you'll have something like
$$\frac{32 + \mbox{ junk }}{8 + \mbox{ stuff }}$$
where junk and stuff all have powers of $n$ on the bottom and so go to zero.
That makes your limit $4$.