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?

Question

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.

Answer 1

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

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I would be more critical towards a sum like

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

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

Answer 2

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.

Answer 3

Be careful with recursively defined sums. $S=1+2S$ doesn't solve for -1 by accident. ie:

$S=1+2S. (1-2)S=1. S=-1.$

The thing is that $S$ is not the sum! $S = Sum(S) + Tail(S)$ in a recursive sequence. So, $S - Tail(S) = Sum(S)$. That's where the negative values come from. $S = 1 + 2 + 4 + 8 + \cdots + 2^n + 2^{n+1}S$. It's important to not stop at "$\cdots$" and throw the tail away.

$(-1) = 1 + 2 + 4 + 8 + 16(-1).$

Notice this:

$S - 2^{n+1}S = 1 + 2 + 4 + \cdots + 2^n.$

There's no paradox at all. The $S=-1$ is why you can now solve for $Sum(S) = 2^{n+1}-1$, and this is a general phenomenon. S is not the sum! $Sum(S)$ is actually $(S - Tail(S))$.