Given a set of numbers $S=\{x_1,\dotsc,x_{|S|}\}$, where $|S|$ is the size of the set, what would be the appropriate notation for the sum of this set of numbers? Is it
$$\sum_{x_i \in S} x_i \qquad\text{or}\qquad \sum_{i=1}^{|S|} x_i$$ or something else?
I'd write either $\displaystyle\quad\sum_{i=1}^{|S|} x_i\quad$ or $\displaystyle\quad\sum_{x\in S} x$.
If the second form is used, then the subscript is just clutter.
Some mathematicians (perhaps especially set theorists?) might write $\displaystyle \sum S$. This would parallel the way set theorists write $\displaystyle \bigcup S$ where others might write $\displaystyle \bigcup_{x\in S} x$.
Both expressions are acceptable with the second being more usual in this context.
The expression $$\sum_{x \in S} x$$ is more common when $S$ is implicitly defined, e.g., when one is summing over all prime numbers. The expression $$\sum_{i =1}^{|S|} x_i$$ would be more common here because you are explicitly given the list of elements of the set $S$.
You can use sigma notation with index:
$$ \sum_{i=1}^{|S|} x_i $$
Alternatively, index may be assigned an explicit set indexes:
$$ \sum_{i \in \langle 1,|S| \rangle} x_i $$
Or go without indexes, like that:
$$ \sum_{x \in S} x $$
There is also other notation when there's a set of numbers that meet specific condition:
$$ \sum_{ \{{ \omega: a\omega^3+b\omega^2+c\omega = 0 \}} } \omega $$
The example above represents a sum of the roots of the $ax^3+bx^2+cx$ expression.
Say I had a set A, under an operation with the properties of $+$, then $$\sum_{i\in A} x_i$$ is how I write it.
