Edited:
- updated sequence notation for use of parentheses instead of angle brackets to reduce ambiguity
- updated starting index of example sequence to be 1 (since final subscript is $n$)
I want to assign a sequence, say $(x_1, x_2, x_3,...,x_{n-2},x_{n-1},x_n)$, to a variable.
In set theory, sets are usually assigned to capital letters, like so: $$Let\ X := \{a,b,c\}.$$
What is standard notation for a variable to which a sequence is assigned?
Notation should account for finite, infinite and bi-infinite sequences.
- i.e. $(x_{1}, x_{2}, x_{3},...,x_{n-2},x_{n-1},x_{n})$
- i.e. $(x_{1}, x_{2}, x_{3},...)$
- i.e. $(x_{-1}, x_{-2}, x_{-3},...)$
- i.e. $(...,x_{-2},x_{-1},x_{0},x_{1},x_{2},...)$
Is $n$ fixed or variates? If it is fixed you usually denote this as subset $A^n$ with $\forall i, \, x_i \in A$, then you just use a lowercase $x = (x_1,\dots,x_n)$.
In some cases like numerical calculus or matricial computations, you can prefer the notation
$$X = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \in \mathbb{R}^n$$ Or any other field, but I personally use $x$ in lowercases all the time to avoid confusing matrices with vectors.
If your $n$ variates, you have a sequence in the classical sense, then better use: $$x = (x_k)_{k \in \mathbb{N}}$$ Note that this notation can also be useful for fixed $n$: $$x = (x_k)_{k \leq n}$$
Also the notation using "$\langle$" and "$\rangle$" is confusing, better use parenthesis or brackets