$\sum^n_{i = n+1} 1$ is $0$ or underfined or identity element for the operation it's in? Let's say you have the $\sum^n_{i = n+1} 1$. The things to add don't exist, because $n+1 > n$. What do you do then?
Do you count it as $0$? Because $0$ is the identity element for addition? What if it was multiplying from $n+1$ to $n$? Would that be $1$?
Should you take the identity element for the operation that takes no arguments? Or should  you take the identity element for the operation that the sum/product is used in? Example:
$$3\cdot \sum^n_{i = n+1} 1 = 3\cdot 0,$$ 
because the identity of additon is $0$ or 
$$3\cdot \sum^n_{i = n+1} 1 = 3\cdot 1,$$
because the identity of multiplication is $1$?
Is there such a thing as undefined in math? How is the result of operations on $0$ elements that require more defined?
Thanks!

An 'empty sum' such as the one you're talking about is defined as 0, because as you say 0 is the additive identity. This is just a convention, though a useful one.
The fact that you then multiply this empty sum by something doesn't change this. The sum still evaluates to 0, which if you multiply by 3 you get 0.
