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?

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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