TITLE: How acceptable is an ellipsis "..." in formal mathematics?
QUESTION [24 upvotes]: The ellipsis "$...$" sometimes seems to be seen as a bit informal. Its use is often justified for cases "where the intention or meaning is clear". And of course, one can go arbitrarily far with nitpicking, intentionally misunderstanding a notation, or suggesting ambiguities. In many cases, the intention really is clear. But for me, it still looks like the writer was handwavingly saying: "Yeah, and so on, y'know what I mean". Usually, the ellipsis can easily be replaced with a more rigorous notation - often involving some sort of indexing over $\mathbb{N}$. And I wonder why this is often not done. 

So my question is:
How acceptable is an ellipsis "$...$" in formal mathematics? 
Of course, this does not refer to textbooks where the natural numbers are introduced as "$\{1,2,3,4,...\}$". It rather refers to mathematical research, or as a specific example: A paper about a proof where the correctness of the proof crucially depends on the right interpretation of an ellipsis, even if it is only used in a basic definition of something "trivial and obvious" like "$a_1 + ... + a_n$". 
How far should one go with trying to avoid the use of the ellipsis, in order to not be confronted with the possible ambiguities or lack of rigour?

I found two questions that are related to this one:

I've been told that the use of ellipsis in "$S = x_1 + x_2 + x_3 + x_4 + \dots$" is ambiguous and meaningless. Is it?
More rigorous notation than "ellipsis" for "and so forth?"

They refer to a particular use of ellipsis "$...$", and how to replace it with a more rigorous notation. Further search reveals attempts to formalize the ellipsis - for example, Proofs About Lists Using Ellipsis (A. Bundy, J. Richardson) states

A  notation  often  used  in  informal  mathematical  proofs  is  ellipsis  (the  dots  in $a_1 + ... + a_n$)
...
The first problem in formalising ellipsis is its inherent ambiguity. The reader of a  formula  containing  ellipsis  has  to  induce  a  pattern  from  the  expressions on either side of the dots. [...] One can try todisambiguate ellipsis by putting in more context [...] but some ambiguity will always remain. More importantly, it is hard to see how we can ensure that a “proof” is in fact a proof unless it can be expressed in an unambiguous internal representation

But this refers to a very specific context, and not to how acceptable the ellipsis is in proofs and definitions in general.

REPLY [10 votes]: The following ten aspects are relevant concerning the question:
1. If there is (almost) no ambiguity for an informed reader and if it increases the readability, there is no problem with using an ellipsis.
...
10. If you can replace an ellipsis by a more rigorous notation without increasing the complexity / decreasing the readability, then you should do so.