Convergent Sequence

A Sequence $S_n$ converges to the limit $S$

$$\lim_{n\to \infty} S_n = S$$

if, for any $\epsilon > 0$, there exists an $N$ such that $\vert S_n-S\vert<\epsilon$ for $n>N$. If $S_n$ does not converge, it is said to Diverge. Every bounded Monotonic Sequence converges. Every unbounded Sequence diverges. This condition can also be written as

$$\overline{\lim_{n\to \infty}} S_n = \underline{\lim_{n\to \infty}} S_n = S.$$

See also Conditional Convergence, Strong Convergence, Weak Convergence