Infinite limit, divergent and indeterminate sequences

Beyond the convergent case, a sequence may grow without bound or fail to settle down at all.

Definition — Infinite limit

$a_n\to+\infty$ if for every $M>0$ there exists $n_0$ such that $a_n > M$ for every $n\ge n_0$. Similarly $a_n\to-\infty$. A sequence with an infinite limit is said to be divergent.

If instead $(a_n)$ admits no limit (neither finite nor infinite) it is said to be indeterminate (or oscillating).

To sum up, a sequence can be of three types: convergent (finite limit), divergent (limit $\pm\infty$) or indeterminate (no limit).

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Topics: Sequences
Concepts: Limit of a sequence · Divergent sequence · Indeterminate sequence