$1,2,\cdots,n$ or $1,\cdots,n$?

Question

In mathematical writing, when should one write "$1,2,\cdots,n$" and when should one write "$1,\cdots,n$"?

It seems to me that writing $1,2,\cdots,n$ is actually wrong if $n=1$ is allowed.

When writing a paper or a thesis, should the author be consistent in the choice? That is, should he stick to one of the two expressions all the time, unless for some very good reasons to choose the other occasionally?

Answer 1

1,2,⋯,n is a convention to make it clear that the list is of positive integers. The opening items in a list should make its iterative pattern reasonably clear, and the sequence should be accompanied by a verbal intension to clarify any pattern that's not obvious.

1,2,⋯,n shouldn't be wrong if n=1 is allowed, only if n=2 is not allowed.

Answer 2

There are two two things to consider:

• clarity,
• consistency.

For example (it's my own experirnce based on reading lots of research papers):

• The context is the king, e.g. non-consistent formatting could be justified if you are reproducing exactly a formula from some other well-known paper (that is, if, for whatever reason, you are reproducing their formatting too).
• Try to keep all such enumerations either $0$-based or $1$-based, never mix the two (if necessary, add $+1$ or $-1$).
• If frequent enough, define $[n]=\{0,1,\ldots,n-1\}$ (or starting from $1$, see above).
• I prefer $1,\ldots,n$ to $1,2,\ldots,n$, because it's shorter.
• However, if there are also more complicated expressions or other sequences, like $1,2,4,\ldots,2^{n-1}$, then I write $0,1,2,\ldots,n-1$ (enough terms to avoid confusion).
• I prefer \ldots to \cdots, that is $0,1,\ldots,n-1$ to $0,1,\cdots,n-1$.
• For more complicated expressions it's beneficial to the reader to include the two last terms, e.g. $1,2,3,\ldots,n-1,n$.
• You should cover the edge cases (e.g. $n=1$) by text or an explicit formula.
• Even if there are some conventions like $\sum_{i=k}^{k-1}\operatorname{whatever}(i)=0$, which go nicely with $1,2,\ldots,n$ being an empty sequence for $n=0$, always, always, explain them (perhaps in the footnote if you don't want to break the thought).

• For infinite sequences include enough terms to make clear what it is, in case of more than arithmetic or geometric sequences use additional clues, e.g. text, middle term or whole formula, e.g.:

  • $0,0,\ldots$, not $0,\ldots$;
  • $1,2,3,\ldots$ if $1,2,4,\ldots$ also happens;
  • $0!, 1!, 2!, \ldots$ rather than $1,1,2,6,\ldots$;
  • $\frac{1\cdot2}{2},\frac{2\cdot3}{2},\frac{3\cdot4}{2},\ldots$ or $1,3,6,\ldots,\frac{i(i+1)}{2},\ldots$ depending on your style;
  • $f(0),f(1),f(2),\ldots$ for anything more complicated.

I hope this helps $\ddot\smile$