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

Question

I know that the notation isn't very important, but I am curious. If I have the following alternating series: $$ \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} $$ Which of the following notations is the most correct (or the best) for represent the previous series?

\begin{align} &\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}+\cdots \tag{1} \\[6pt] &\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots \tag{2} \\[6pt] &\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\pm\cdots \tag{3} \\[6pt] &\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}+-\cdots \tag{4} \\[6pt] &\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\cdots \tag{5} \\ \end{align}

I think that is the equation $(3)$, but I am not sure.

Answer 1

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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 $$\frac{1}{0!}−\frac{1}{1!}+\frac{1}{2!}−\frac{1}{3!}+\frac{1}{4!}−\cdots+\frac{(−1)^n}{n!}+\cdots$$

Answer 2

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.