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Convergent Sequence

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

\begin{displaymath}
\lim_{n\to \infty} S_n = S
\end{displaymath}

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

\begin{displaymath}
\overline{\lim_{n\to \infty}} S_n = \underline{\lim_{n\to \infty}} S_n = S.
\end{displaymath}

See also Conditional Convergence, Strong Convergence, Weak Convergence




© 1996-9 Eric W. Weisstein
1999-05-26