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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. $(...,x_{-2},x_{-1},x_{0},x_{1},x_{2},...)$
Jacob Barnard
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    If this is a finite sequence it can just be viewed as a vector $x=(x_1, ..., x_n)$. If you want you can make it have countably infinite dimensions $x=(x_1, x_2, x_3, ...)$. – Michael Jul 03 '23 at 15:04
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    There's even more notations to choose from! I have some affection for an old "underline" notation of the bi-infinite sequences which are the elements of a Bernouli shift: $$\underline{x} = (\ldots,x_{-2},x_{-1},x_0,x_1,x_2,\ldots) $$ – Lee Mosher Jul 03 '23 at 15:23

1 Answers1

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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

julio_es_sui_glace
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    +1 : I gave essentially the same answer in a comment a few seconds before. I agree that the inner product style notation looks confusing. – Michael Jul 03 '23 at 15:07
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    @Michael sorry I was typing my answer :) – julio_es_sui_glace Jul 03 '23 at 15:08
  • So would you disagree with the accepted answer here? https://math.stackexchange.com/questions/115978/notation-for-sequences – Jacob Barnard Jul 03 '23 at 19:45
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    Yes absolutely, it adds ambiguity, and we want to avoid that. Also the parenthesis notation is almost universal today – julio_es_sui_glace Jul 03 '23 at 19:50
  • @julio_es_sui_glace - Understood. I cracked open an old discrete math textbook. It favors parentheses over angle brackets as well. However, it almost always uses the word "series" when employing the notation, not "sequence." Does that impact your answer here? (I'd like to leave the question open a bit longer either way to allow more feedback.) Thanks! – Jacob Barnard Jul 03 '23 at 19:53
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    Old books use often various and not universal notations. I also know that they are some books that use series instead of sequence (old ones too), but maybe they are talking about the series associated to this sequence? – julio_es_sui_glace Jul 03 '23 at 19:56