What does it mean when there's an "element of" under a sigma summation

Question

$\sum\limits_{a_1 \in \{1, x\}} a_1$

The context is that it's used to expand $(1 + x)^n$ into $(1 + x)(1 + x)..$ $=$ $(\sum\limits_{a_1 \in \{1, x\}} a_1)$ $(\sum\limits_{a_2 \in \{1, x\}} a_2)..$

What's the definition for this summation notation? Thanks!

edit: Curly brackets instead of square brackets under the sigma!

Answer 1

$\sum\limits_{a_1 \in \{1, x\}} a_1$ simply means $1+x.$

$a_1$ runs through all the elements of the set $\{1,x\}$, which are $1$ and $x$, and $\sum$ means to add them.

Answer 2

$\sum_{a\in S}f(a)$ means "substitute each member of the set $S$ into the function $f$ and add the results together"; $\{1,x\}$ is a set. The more common notation $\sum_{k=a}^bf(k)$ is equivalent to $\sum_{k\in\{a,\dots,b\}}f(k)$.