The Flint Hills series, is the series $$\sum_{n=1}^\infty\frac{1}{n^3\sin^2(n)},$$ and it's an open problem as to whether the series converges. From the proof of Corollary 4 of this paper, it seems that it's also unknown if $$\lim_{n\to\infty}\frac{1}{n^3\sin^2(n)}$$ converges. Generally speaking (at least when it comes to problems one encounters in a typical calculus/analysis course), given a sequence $(a_n)$ it is easier to check the convergence of $$\lim_{n\to\infty}a_n,$$ than the convergence of $$\sum_{n=1}^\infty a_n.$$
Why then is the problem of determining if the series $$\sum_{n=1}^\infty\frac{1}{n^3\sin^2(n)}$$ converges a more pressing matter than determining if $$\lim_{n\to\infty}\frac{1}{n^3\sin^2(n)}$$ converges?