What is the notation for alternating series with "$\cdots$"?

I would use $(5)$ or maybe $(2)$, but I don't think it is very important. What is important is to show enough terms that the pattern is obvious. I would not rely on the trailing sign to convey that information, which is why I would use $(5)$. I think $(5)$ is sufficient, but I might show the next term before I went to dots.

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(2) is most common. (See, for instance, some examples in Wikipedia's "Taylor series" entry.) So long as you've included enough terms to establish the pattern, you're fine. In cases where the pattern may not be entirely clear, the sigma notation is best; alternatively, you can provide this hybrid form $$\frac{1}{0!}−\frac{1}{1!}+\frac{1}{2!}−\frac{1}{3!}+\frac{1}{4!}−\cdots+\frac{(−1)^n}{n!}+\cdots$$