Introduction to sequences

In this lesson, we will talk about what sequences are and how to formally write them. Then we will learn how to write the terms out of the sequences when given the general term. We will also learn how to write the general term when given a sequence. After learning the notations of sequences, we will take a look at the limits of sequences. Then we will take a look at some definitions and properties which will help us take the limits of complicating sequences. These theorems include the squeeze theorem, absolute value sequences, and geometric sequences.

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.