In math, we sometimes talk about the parity of a number - that is, what Wikipedia succinctly describes as " an integer's inclusion in one of two categories: even or odd." Is there a similar word for convergence / divergence that would allow you to say "Find the X (convergence / divergence) of the sequence"?
What you're referring to is the behaviour of the 'far-reaches' of sequence. In other words, the sequence's asymptotic behaviour.
Here's a literal rendering of the term asymptote as well as the more abstract application of the concept in asymptotic analysis:
Although analysing asymptotic behaviour provides information on whether a sequence converges or diverges, it typically goes beyond just convergence and divergence, looking for a function that essentially ignores complex behaviour that occurs in early parts of the sequence. It considers the question:
how is the system behaving "after a long time?" - Contemporary Calculus, section 4.6
If you want to restrict the consideration to just convergence and divergence, you can rephrase the question to "does the sequence converge?".
Not a word, but a notation that can be unpacked easily. Two actually.
It is best known for characterizing the divergence of high growth rate behaviors such as the difficulty of solving a problem computationally as the size of the input grows. But it is equally suited to describing convergent series and is used that way in several specialties.
edit fixed link. https://en.wikipedia.org/wiki/Big_O_in_probability_notation