Sum without an index Is $\sum a$ a customary (standard) shorthand for $\sum_{i\in\operatorname{dom}a} a_i$, where $a$ is an indexed family of say integers?
 A: You will sometimes see it used that way, but in my view it’s a dismally poor abuse of notation. At the very least the index should appear somewhere in the expression: $\sum_ia_i$ is fine, given a reasonable context, or even $\sum a_i$, but $\sum a$ is at best annoying and at worst confusing, especially since $\sum_{k=1}^na$ has the completely different unambiguous meaning $na$.
Added: It occurs to me belatedly that there is one context in which I would not at all object to the notation $\sum a$: if $a$ is a finite set of real numbers, say, $\sum a$ is perfectly acceptable shorthand for $\sum\{x:x\in a\}$, just as in set theory $\bigcup a$ is unambiguously $\bigcup\{x:x\in a\}$ if $a$ is a set of sets.
A: Yes. IMO there's not much of a problem with it: in a Haskell-ish pseudo-lambda-calculus-notation
$$\begin{align}
  &\Sigma\ ::\ (J\text{ countable}, S\text{ additive})\Rightarrow\ (J \to S) \to S \\
  &\Sigma f = \underbrace{f(j_1) + f(j_2) + \ldots}_{\text{all }j_k\in A}
\end{align}$$
with the more common general notation just being shorthand
$$
  \sum_{i\in I}a_i := \Sigma\bigl(\lambda i.\ a_i\,\chi_I(i)\bigr)
$$
where $\chi_I(i)=1$ for $i\in I$ and $0$ otherwise.
A: Often, yes. The $a_i$ need not be integers, and the index set can also be different from integers - it's usually understood from context. The same goes for products, $\prod a_i$.
