sequences and series - Does $a_n = frac{1}{n}$ converges even though $sum frac{1}{n}$ diverges?

I've been studying sequences and series recently. As I understood, the sequence convergence is determined whether the sequence has a limit value. Now, in this example

$$a_n = \frac{3n^2 - 5n + 7}{3n^3 - 5n + 7}$$

I get $\lim_{n\to \infty} \frac{1}{n} = 0$, which means this sequence converges. What confuses me is that it is known that series $\sum \frac{1}{n}$ diverges. My question is, is it possible that $\frac{1}{n}$ converges when working with sequences, but diverges when working with series? Or it diverges in both cases?

Thank you in advance