Introduction to sequences - Sequence and Series

Notes:

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.