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?

What else can the definition of the ellipsis symbol, "$$\dots$$", mean in this context?

$$S = x_1 + x_2 + x_3 + x_4 + \dots$$

All I can see is that you have an infinite sum of $$x$$s, where the first one is $$x_1$$, the next is $$x_2$$, then $$x_3$$, and so on forever, for as many natural numbers as exist and in order. But, for some reason, I am being told that such a definition is ambiguous and meaningless compared to formal mathematics.

• What if the next term is $x_5$, for example? I think that's what the person who told you that means. It's not as precise as writing $S=\sum_{n=1}^\infty x_n$ – Jakobian Oct 4 '18 at 18:13
• The expression $x_1+x_2+x_3+x_4+\dots$ generally is taken to mean $\lim\limits_{n\to\infty}\sum\limits_{i=1}^n x_i$. On the other hand, $x_1+x_2+x_3+\dots+x_n$ is generally taken to mean $\sum\limits_{i=1}^n x_i$. – JMoravitz Oct 4 '18 at 18:14
• @ Jakobian, sure the next term could ba anything, even $p_1g$, but a person would really have to desire to misunderstand to make such an objection. – Ivan Hieno Oct 4 '18 at 18:21
• The only scenario that usually comes up that I would say is ambiguous (only in that people don't follow standard convention) is that $\lim\limits_{n\to\infty}\sum\limits_{i=1}^n i = \infty$ while some people will write something like "$1+2+3+\dots = -\frac{1}{12}$" such as here. It is important to note that people who write that are going against standard convention. Most mathematicians will agree that the only valid answer to evaluating $1+2+3+\dots$ is $\infty$. – JMoravitz Oct 4 '18 at 18:22
• What they intend with the $-\frac{1}{12}$ interpretation is that $1+2+3+\dots$ is in effect shorthand for the analytic continuation of the zeta function. – JMoravitz Oct 4 '18 at 18:23

Stating ambiguity of this summation is bad faith. The pattern is obvious and the ellipsis clearly indicates an unlimited sequence of terms.

I would be more critical towards a sum like

$$1+2+4+\cdots\ ?$$

In common practice, if the first few terms/indexes (as little as $$3$$) follow an arithmetic progression, it can be considered implied.

To give you an idea of what can go wrong with arbitrary entries in the summation the following example shows these complications. Begin with $$S=1 + 2 + 4 + 8 + ...$$ and so $$S=1+2(1+2+4+8+...)$$ and so $$S=1+2S$$ giving us finally that $$S=-1$$ and we now have a sum of positive numbers equal to a negative number, which is absurd. These complications must be addressed in order to deal with sums of this kind meaningfully.

• This "problem" has nothing to do with the ellipsis notation. – Yves Daoust Oct 1 '19 at 21:04