Notation: What is the scope of a sum?

I would interpret $\sum_{i=1}^2 x_i + y$ as $x_1 + x_2 + y$, but I would interpret $\sum_{i=1}^2 x_i + y_i$ as $x_1 + y_1 + x_2 + y_2$. I realize this is a little inconsistent. Should the latter be written as $\sum_{i=1}^2 (x_i + y_i)$?

Or, in other words, does the sum operator have precedence over + and - ?

Solution 1:

It should be written with parentheses to avoid ambiguity, yes. If you think about the "sum" symbol as a function, it makes sense:

$$ \sum(\cdot) $$

This is a function which takes a list $\{x_1,x_2,\ldots\}$ of numbers (or other mathematical objects which you might want to add), and adds them in order. This list could be finite or infinite: the "sum" function figures out how long the list is, and adjusts its indexing accordingly (i.e. if there are 10 things in the list, your index will go from $1$ to $10$).

If you want to add a sequence which is itself the addition of two sequences, like your example of $\{x_1+y_1,x_2+y_2\}$, you'll need to drop the whole sequence into the function:

$$ \sum(\{x_1+y_1,x_2+y_2\})=\sum_{i=1}^2(x_i+y_i) $$

For finite sums of numbers, we always have the property that

$$ \sum(\{x_1,x_2,\ldots\}+\{y_1,y_2,\ldots\})=\sum(\{x_1,x_2,\ldots\})+\sum(\{y_1,y_2,\ldots\}) $$ however the following would have a different interpretation:

$$ \sum(\{x_1,x_2,\ldots\})+\{y_1,y_2,\ldots\} $$

Hence the parentheses!

Interesting, yet more advanced side note: "breaking up" a sum doesn't always work if the sequences are infinite!