Limit of a sequence

The value $a_n$ approaches as $n\to\infty$ ; a sequence CONVERGES if this limit exists, DIVERGES otherwise.

A sequence $\{a_n\}$ converges to $L$ (written $\lim_{n\to\infty}a_n=L$ ) if $a_n$ gets arbitrarily close to $L$ as $n$ grows without bound — the same intuition as a function limit at infinity, restricted to integer inputs. A sequence that doesn't approach any single finite value diverges, whether by growing unboundedly or by oscillating without settling.

Examples

$a_n=\dfrac1n\to0$ as $n\to\infty$ — converges.
$a_n=(-1)^n$ oscillates between $-1$ and $1$ forever — diverges (never settles on one value).