My professor refers to it as "con-dive behavior", but I'm almost certain that that's not an actual term. For example, he might ask: "what is the condive behavior of the harmonic series?", to which we would answer "divergent".
Is there a legitimate word/terminology that has the same usage as "condive behavior" in that example?
The professor, perhaps, is using short-term "condive behaviour" to mean "convergence-divergence behaviour". But this is not a world-wide accepted acronym.
Another technical term you might like to use is: "limiting behaviour" of the series (as $n\to\infty$).