Sequences

Any ordered list of numbers

Can be represented by a function using a general term
${a_{n}}_{n = 1}^{\infty}$

even = 2n
odd = 2n+1/2n-1

Most notable(well-known) sequences are Discrete Sequences

Limits of Sequences

lim_{n \to \infty} a_{n} = L

For every, $ϵ > 0$ , there exists N>0 such that if $n \geq N$ , then $a_{n} - L < ϵ$

Limit laws applied to sequences

$lim_{n \to \infty} a_{n}^{p} = (lim_{n \to \infty} a_{n})^{p}$
$lim_{i \to \infty} a_{n}$ and $f$ is continuous at $L$ $\to$ $lim_{n \to \infty} f (a_{n}) = f (L)$
$lim_{n \to \infty} | a_{n} | = 0 \to lim_{n \to \infty} a_{n} = 0$
$lim_{n \to \infty} a_{2 n} = L$ and $lim_{n \to \infty} a_{2 n + 1} = L \to$ $a_{n}$ converges to $L$
L'Hopital's Rule
Squeeze Theorem

Recursive Sequences

a_{n + 1} = f (a_{n}, a_{n - 1}, a_{n - 2}, \dots)

Sequences like the fibonnaci or the geometric sequence where there is one common ratio between all the terms $r$

$a_{n + 1} = r a_{n}$

Qualitative information is used to determine the limits of a recursive sequence

$a_{n} < a_{n + 1}$ for all $n \geq 1$ $\to {a_{n}}$ is monotone increasing
$a_{n} > a_{n + 1}$ for all $n \geq 1$ $\to {a_{n}}$ is monotone decreasing

Which can be extracted using inequalities and calculus

derivatives
Discrete Sequences#Induction

Bounds

${a_{n}}$ is bounded above if $\exists M \in R$ such that $\forall$ n, $M \geq a_{n}$
${a_{n}}$ is bounded below if $\exists m \in R$ such that $\forall$ n, $m \leq a_{n}$

Monotone Increasing/Decreasing Sequence Theorem

If a sequence is monotone increasing/decreasing and they have a bound below or above, $L \leq M$ / $L \geq M$ (inc/dec)