Answer
See below.
Work Step by Step
$\text{ What is an infinite sequence? }$
An infinite sequence of numbers is a function whose domain is the set of positive integers.
We denote the sequence $\{f(n)|n\in \mathbb{N}\}$ as $\{a_{n}\}$ where $f(n)=a_{n}.$
In simple terms, a list of numbers for which the first is $a_{1}$, second is $a_{2}$, nth is $a_{n}$ and there is a term corresponding to every natural number (which is why it is an infinite sequence).
$\text{What does it mean for such a sequence to converge? To diverge?}$
The sequence $\{a_{n}\}$ converges to the number $L$ if for every $\epsilon\gt 0$
there exists an integer $N$ such that for all $n\gt N$ we have $ |a_{n}-L| \lt\epsilon.$
If no such number $L$ exists, then the sequence $\{a_{n}\}$ diverges.
If $\{a_{n}\}$ converges to $L$, we write $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$
and call $L$ the $limit$ of the sequence.
$\text{Give examples.}$
Examples of convergent sequences:
$a_{n}=1\qquad \left\{1,1,1,\right\}$ converges to 1
$a_{n}=\displaystyle \frac{1}{n}\qquad \left\{1,1/2,1/3,\right\}$ converges to 0
Examples of divergent sequences
$a_{n}=n\qquad \left\{1,2,3,\right\} \quad$
$a_{n}=(-1)^{n}\qquad \left\{-1,1.-1,1,\right\}$
