Why sigma notation? Repeated union is written as:
$$\bigcup_{i=0}^na_i$$
Repeated logical conjunction is:
$$\bigwedge_{i=0}^na_i$$
Etc.
So why isn't repeated addition:
$$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large a_i}$$
Why use Sigma and Pi for sums and products? Everything else is just a bigger version of the symbol.
 A: (This is too long for a comment so it's being posted as an answer)
Good question. I think the display may appear visually confusing if you have to add a few summations, e.g.
$$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large a_i}\operatorname{\huge+}\operatorname{\huge+}\limits_{i=0}^n{}^{\Large b_i}\operatorname{\huge+}\operatorname{\huge+}\limits_{i=0}^n{}^{\Large c_i}$$
(the addition sign has been exaggerated to illustrate the point)
This is much clearer:
$$\sum_{i=0}^na_i\operatorname{\large+}\sum_{i=0}^n b_i\operatorname{\large+}\sum_{i=0}^n c_i$$
A: $\sum$ is the notation used for summation because sigma ($\sum$) is the Greek letter for $s$.
Note that $\oplus$ is the notation used for direct product of algebraic objects (such as groups, vector spaces, modules, etc). This notation seems to be close the the notation you have suggested. It also separates the operator from the operand more cleanly.
For instance, the following notation would be more cumbersome:
\begin{align}
\operatorname{\huge+}\limits_{a+b=c}{x^{a} y^{b}}
\end{align}
than this notation:
\begin{align}
\bigoplus_{a+b=c}{x^{a} y^{b}}.
\end{align}
I should note that most of what I have said is due to reasonable guesses.
