divergence
If convergence is a sequence settling down to one place, divergence is the catch-all name for everything else: the sequence simply refuses to home in on a single finite value. This can happen in different ways — by exploding to infinity, by oscillating forever, or by wandering chaotically without ever committing.
Formally, a sequence diverges if it does not converge — that is, there is no finite real number L with a_n -> L. Divergence is therefore defined negatively, as the failure of the convergence condition. Two important sub-cases are worth naming: a sequence may diverge to infinity (growing past every bound), or it may diverge by oscillation (bouncing between values that never reconcile).
It is a common slip to think a non-convergent sequence must run off to infinity. Not so: the bounded sequence (-1)^n diverges while staying trapped in [-1, 1]. Divergence is the absence of a limit, nothing more, and proving divergence usually means exhibiting two subsequences with different limits, or showing the terms escape every fixed bound.
a_n = (-1)^n diverges by oscillation: its even terms equal 1 and its odd terms equal -1, so two subsequences have limits 1 and -1. Meanwhile a_n = n diverges to infinity. Both are divergent for different reasons.
Divergence includes both unbounded growth and bounded oscillation.