Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

4 Answers 4

up vote 6 down vote accepted

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.

share|improve this answer
    
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.

share|improve this answer
    
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.

share|improve this answer

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 $x \mapsto f(x)$).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.