# Sequence Notation — Which brackets to use?

I'm teaching sequences at the moment. I've always put sequences in round brackets, for example $(1,2,3,4,5)$ is a sequence whose first member is $1$, whose second member is $2$, and so on. I've also always used round brackets to define a sequence in the following way: "Consider the sequence $(a_n)$ where $a_k = k^2+1$ for all $k \ge 1$." I would like to know if this is in standard usage.

On Wikipedia, they use the notation $\{a_n\}$ for a sequence. I thought "curly brackets" were reserved for sets where order in unimportant, e.g. the sets $\{1,2\}$ and $\{2,1\}$ are the same set. While in a sequence, the order does matter, e.g. $(1,2) \neq (2,1)$. Just like the points in the $xy$-plane differ.

To compound it even further, the course text does not use any brackets at all. For example, they say "Find the next term in the geometric sequence $1, 2, 4, 8,\ldots$

Of course I realise that we can use any notation we choose, provided we define it beforehand, but I'm interested to hear people's preferences and their own experiences.

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I use $(a_n : n \in \mathbb{N})$ and I suppose $(a_n)_{n \in \mathbb{N}}$ would also be okay. I think $\langle a_n : n \in \mathbb{N}\rangle$ is good too, but I think $\langle a_n \rangle_{n \in \mathbb{N}}$ is fairly uncommon. I agree that it's best not to use curly braces except in contexts where it really is okay to forget about the order.

Personally I don't feel comfortable writing just $(a_n)$ (leaving $n$ as a free variable) for an infinite sequence; $a_n$ is a number, so $(a_n)$ is just a sequence of length one.

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That's a very good point that you make about $(a_n)$. –  Fly by Night Mar 7 '13 at 17:40

I much prefer angle brackets: $\langle 1,2,4,8,\dots\rangle=\langle 2^k:k\in\Bbb N\rangle$. Parentheses are a distant second choice: they already have too much work to do. I consider curly braces utterly inappropriate: $\{2^k:k\in\Bbb N\}$ is a set of integers, not a sequence.

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Interesting! I've always used angled brackets for things like group representations like $D_3 := \langle x,y : x^3=y, xy=yx^{-1}\rangle$. I don't think I've ever seen angled brackets around sequences. I take your point about brackets having enough work to do. Would you use $\langle 1,2\rangle$ to denote a point in the plane? –  Fly by Night Mar 7 '13 at 17:22
@FlybyNight: Yes; I routinely do so. I think that the convention is most common amongst those with set-theoretic backgrounds. –  Brian M. Scott Mar 7 '13 at 17:23
That's really interesting. Thanks Brian. –  Fly by Night Mar 7 '13 at 17:24
@FlybyNight: You’re welcome. –  Brian M. Scott Mar 7 '13 at 17:25

I mostly use parenthesis or curly braces to denote sequences although with the modification that I put a subscript on. For example:

• Let $(a_n)_{n\geq 1}$ be a sequence of real numbers.
• Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers.
• Let $(a_n)_{n\geq 1}$ be the sequence given by $$a_n=\frac1n,\quad n\geq 1.$$

However, I appreciate the point made by Brian M. Scott in the other answer.

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If I need to make it clear, I use $n \mapsto x_n$, for example. Other notations are fairly entrenched, so it is probably best that your students learn them.

While rather sloppy, I sometimes write 'the sequence $x_n$...', just as some peope write 'the function $f(x)$...' (meaning the function $f$, or $x \mapsto f(x)$).

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