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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.

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  • $\begingroup$ 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 $S=\sum_{n=1}^\infty x_n$ $\endgroup$ – Jakobian Oct 4 '18 at 18:13
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    $\begingroup$ 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 $\sum\limits_{i=1}^n x_i$. $\endgroup$ – JMoravitz Oct 4 '18 at 18:14
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    $\begingroup$ @ 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. $\endgroup$ – Ivan Hieno Oct 4 '18 at 18:21
  • $\begingroup$ 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 $\infty$. $\endgroup$ – JMoravitz Oct 4 '18 at 18:22
  • $\begingroup$ 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. $\endgroup$ – JMoravitz Oct 4 '18 at 18:23
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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.

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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.

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

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