The [[Sequence]] $\set{a_n}$ has the limit $L$, written as: $ \huge\lim_{n\to\infty} a_{n} = L $ If the limit $\lim_{n\to\infty} a_n$ exists (as a [[Real Numbers|Real Number]]), we say the [[Sequence]] $a_n$ [[Convergent Series|Converges]], else it [[Divergent Series|Diverges]]. #### Precise Definition $\lim_{n\to\infty} a_{n} = L$ if for every $\epsilon > 0$, there is an integer $N$ such that $\left| a_{n}-L \right| < \epsilon$ ### Thereom(s) Consider a sequence $\set{a_{n}}$ such that $a_{n} = f(n)$ for all $n \ge 1$. If there exists a real number $L$ such that $\lim_{{x\to\infty}}f(x)=L$, then $\set{a_{n}}$ convergees and $\lim_{a_{n}\to\infty}=L$.