Sequences

Sequences $LaTeX: \{a_n\}$ , examples;
recursively defined sequences; examples:
- the sequence of natural numbers $LaTeX: \{1,2,3,\ldots\}$ defined by $LaTeX: a_1=1, a_n=a_{n-1}+1$ for $LaTeX: $n\geq 2$$ ,
- Fibonacci sequence $LaTeX: \{f_n\}$ defined by , , $LaTeX: f_n=f_{n-1}+f_{n-2}$ for $LaTeX: n \geq 3$ ;
limit of a sequence - motivation, definition, examples;
a useful theorem: if one can construct a continuous function such that for all $LaTeX: n\geq 1$ , and $LaTeX: \displaystyle{\lim_{x\to\infty} f(x)=L}$ , then $LaTeX: \displaystyle{\lim_{n\to\infty}a_n = L}$ ;
example: $LaTeX: \displaystyle{ \lim_{n\to\infty} \frac{(n-1)^2}{n^2-1} = 1 }$ ;
theorem: for every , $LaTeX: \displaystyle{ \lim_{n\to\infty} \frac{1}{n^r} }$ ;
definition of an infinite limit;
examples;
Limit Laws;
Squeeze Theorem;
Theorem: if $LaTeX: \lim |a_n| = 0$ , then $LaTeX: \lim a_n = 0$ ;
on the other hand, if $LaTeX: \alpha \neq 0$ and $LaTeX: \lim |a_n| = \alpha$ , then $LaTeX: \lim a_n$ may not exist (example: .

[Sec. 11.1]