Definition

Definition

Let A \subseteq \mathbb{N} and S be a set.

If f \colon A \to S is a function, then f is a sequence.

Sequences may be finite or infinite, and their terms may be numbers or functions.

Notation

Let f \colon A \to S be a sequence.

It is common to choose some letter, usually s, to represent the elements of this sequence.

For each k \in A, f(k) is denoted s_k, and f is denoted (s_k)_{k\in A}.

For example, the following is a sequence of Fibbonachi numbers,

\left(1,1,2,3,5,8,13,\dots\right).

We would denote the sequence (F_k)_{k\in \mathbb{N}}, where F_1 = 1, F_2 = 1, F_3 = 2, ect.