Note:

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.