If we say that a sequence converges, it means that the limit of the sequence exists as n → ∞ n\to\infty n→∞. If the limit of the sequence as n → ∞ n\to\infty n→∞ does not exist, we say that the sequence diverges. A sequence always either converges or diverges, there is no other option.

Can a sequence be convergent and divergent?

Every infinite sequence is either convergent or divergent. A convergent sequence has a limit — that is, it approaches a real number. A divergent sequence doesn't have a limit. ... In many cases, however, a sequence diverges — that is, it fails to approach any real number.

Can sequences not diverge or converge?

Every sequence of real numbers either converges or diverges. This is trivial, since divergence means the opposite of convergence. And the sequences that you mentioned diverge.