Sequence

These are just ordered lists of elements. Essentially we talked about Countable Sets, so in the same vein we can have an order to the elements $a_{1}, a_{2}, a_{3}, . . .$ .

Sequence

A sequence is a function $f$ where the domain is just $N$ .

For instance, consider the sequence of numbers ${\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, . . .}$ . What do we mean that the sequence is a function with domain $N$ ? Essentially $1 \mapsto \frac{1}{2}, 2 \mapsto \frac{1}{4}$ , and so on. In general $n \mapsto \frac{1}{2^{n}}$ .

Logically, having either a finite sequence or an infinite sequence, both are countable, which is the key here.

Notation

We write a sequence in any of the following notations:

$(a_{1}, a_{2}, a_{3}, . . .)$
$(a_{n})_{n = 1}^{\infty}$
$(a_{n})_{n \in N}$
$(a_{n})$

From calculus ...

We may start with an index $n$ starting from $0$ or some other integer $k \in Z$ .

Why do we care about sequences? They're important for describing infinite sums, known as Series, Convergence of A Series.

Examples of Sequences

i. $(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots)$
ii. $(\frac{1 + n}{n})_{n = 1}^{\infty}$
iii. $(a_{n})$ where $a_{n} = 2^{n}$ for each $n \in N$
iv. $(x_{n})$ where $x_{1} = 2$ and $x_{n + 1} = \frac{x_{n} + 1}{2}$