1. If a sequence has the limit $L$, then we can say that:

$\lim$

_{n →$\infty$}$a$

_{$n$}$=L$

If the limit is finite, then it is convergent. Otherwise, it is divergent.

2. If the limit of the sequences {$a_n$} and {$b_n$} are finite and $c$ is constant, then we can say that

i) $\lim$

_{n →$\infty$}$(a_n+b_n)=\lim$

_{n →$\infty$}$a_n+$$\lim$

_{n →$\infty$}$b_n$.

ii) $\lim$

_{n →$\infty$}$(a_n-b_n)=\lim$

_{n →$\infty$}$a_n-$$\lim$

_{n →$\infty$}$b_n$.

iii) $\lim$

_{n →$\infty$}$ca_n=c$ $\lim$

_{n →$\infty$}$a_n$.

iv) $\lim$

_{n →$\infty$}$(a_nb_n)=$ $\lim$

_{n →$\infty$}$a_n*$ $\lim$

_{n →$\infty$}$b_n$.

v) $\lim$

_{n →$\infty$}$[a_n$$\div$$b_n]$ $=\lim$

_{n →$\infty$}$a_n$$\div$ $\lim$

_{n →$\infty$}$b_n$$,$$b_n\neq0$.

3. If $a_n\leq c_n\leq b_n$ and $\lim$

_{n →$\infty$}$a_n=$ $\lim$

_{n →$\infty$}$b_n=L$, then $\lim$

_{n →$\infty$}$c_n=L$.

4.if $\lim$

_{n →$\infty$}$|a_n|=0$, then $\lim$

_{n →$\infty$}$a_n=0$ as well.

5. We say that:

Where the sequence {$x^n$} is convergent for -1< $x \leq$ 1, and divergent if $x$ > 1.