convergence
Convergence is the good fate a sequence can have: its terms settle down toward a single finite destination and stay there. Think of a noisy radio signal that gradually tunes in until you hear one clear note — the static dies away and a definite value emerges.
Precisely, a sequence converges if it has a (finite) limit; that is, there exists a real number L with a_n -> L. A sequence that converges is called convergent, and one that does not is called divergent. Note the insistence on finite: a sequence racing off to infinity is, by the standard convention, not convergent even though its behavior is highly regular.
Convergence is a property of the tail of the sequence, not of any finite block of early terms: changing, deleting, or inserting finitely many terms cannot create or destroy convergence, nor change the limit. This is why analysts often say things hold 'for all sufficiently large n' — the first thousand or first billion terms are irrelevant to whether a sequence converges.
The sequence a_n = 1/2^n = (1/2, 1/4, 1/8, ...) converges to 0. Prepending a hundred arbitrary terms in front of it changes none of that: it still converges to 0.
Convergence depends only on the tail, never on finitely many initial terms.
Every convergent sequence is bounded, but the converse fails: (-1)^n is bounded yet does not converge. Boundedness is necessary but not sufficient for convergence.