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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.

Akiva Weinberger
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    A tradition, probably. I don't think that there's any better reason. The sigma notation is older than bigcup notation. – xyzzyz Aug 18 '15 at 15:28
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    Just a guess: someone set that standard and it's stuck since. I think it's most likely that $\Sigma$ was chosen because you are taking a Sum, and the corresponding Greek letter is $\Sigma$, similarly, $\Pi$ for a Product – jameselmore Aug 18 '15 at 15:28
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    Another detail is that sigma notation is older than pi notation Date of introduction of some symbols. I agree with xyzzyx and jameselmore. – 永劫回帰 Aug 18 '15 at 15:35
  • I am sure I have an old book where $A\cdot B = A \cap B$ and $A+B=A \cup B$. – Siminore Aug 18 '15 at 15:48
  • It hasn’t always been true that ‘[e]verything else is just a bigger version of the symbol’. In older books and papers you can find $\sigma_iA_i$ for $\bigcup_iA_i$ and for $\bigvee_iA_i$, and you can find $\prod_iA_i$ for $\bigcap_iA_i$ and for $\bigwedge_iA_i$. – Brian M. Scott Aug 18 '15 at 17:24

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(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:

$$\sum_{i=0}^na_i\operatorname{\large+}\sum_{i=0}^n b_i\operatorname{\large+}\sum_{i=0}^n c_i$$

Hypergeometricx
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    Much the same could be said for $\displaystyle\bigcup_{i=0}^na_i\cup\bigcup_{i=0}^nb_i\cup\bigcup_{i=0}^nc_i$. – Akiva Weinberger Aug 18 '15 at 16:12
  • To be fair, your formula shall read as $$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large a_i}+\operatorname{\huge+}\limits_{i=0}^n{}^{\Large b_i}+\operatorname{\huge+}\limits_{i=0}^n{}^{\Large c_i}$$ which would be a bit more readable. Also pretty likely we'd get used to that if we would not have $\sum$ and $\prod$ for historical reasons. – SBF Aug 27 '15 at 06:59
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    Yes but I did mention the "+" was exaggerated to illustrate the point. Anyway $\sum$ and $\prod$ are pretty cool symbols. They also have convenient "ceilings" and "floors" where the lower and upper limits can rest comfortably. – Hypergeometricx Aug 27 '15 at 07:07
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$\sum$ 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.