Why do we distinguish between infinite cardinalities but not between infinite values? More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$?
Can we distinguish between "countable divergence" and "uncountable divergence"?
I apologize in advance for the "rather naive" question.
 A: Let me just add to Kyle's answer that in cardinal arithmetic you do have a well-defined notion of uncountable sums. Because you're not trying to calculate a real number, rather you're looking for a cardinal.
If you are looking at the sum $\sum_{r\in\Bbb R}1$ as a real-valued sum, then it is indeed a strange thing to write. But looking it at as a cardinal sum, this makes perfect sense, and the cardinal is exactly $2^{\aleph_0}$. Similarly, of course, $\sum_{n\in\Bbb N}1=\aleph_0$ as a cardinal sum, and it's just a divergent series when looking at it as a real-valued sum.
So context is key.
A: So, Cardinal Arithmetic, Natural Arithmetic, and Real Arithmetic are all different things. At least for me, addition in each case means something completely different. 
Cardinal Arithmetic has to do with addition of sets, which is really closer to (disjoint) union modded out by an equivalence relation (i.e. bijection). In this area, we say that $|A| = |B| \iff$ there exists a bijection between the two sets. So, when we say something like $\aleph_0 + \aleph_0 = \aleph_0$, we are really just saying that there exists a bijection between $\aleph_0 \sqcup \aleph_0$ and $\aleph_0$. We say that $\kappa < \mu$ if there exists an injection from $\kappa$ to $\mu$, but none of those injections are surjections. From these definitions, one can show that we have accurately described an equivalence relation in the class of sets. 
Natural Arithmetic, or arithmetic on $\mathbb{N}$ is the arithmetic we learn in school. While we don't learn all the axioms of arithmetic (first of all, because the set of axioms for true arithmetic is not recursive) the basic theory is the pretty much the same as Finite Cardinal Arithmetic. Things change drastically when we consider Non-standard models of arithmetic. In these models of arithmetic, we can add, subtract, multiply and sometimes divide numbers which are larger than all standard natural numbers. We might have a number $\tau$ such that $\tau >n$ for all $n$ in the standard model. Furthermore, $\tau + \tau \neq \tau$. Here, we have that $ \tau <\tau + \tau = 2\tau $ and how you would think Natural Arithmetic might work on very large numbers (with a loose interpretation of large). It should also be stated, however, due to a theorem of Tennenbaum, that none of these arithmetics on non-standard models are computable, i.e. we cannot truly compute anything. 
Finally, Real Arithmetic has two separate meanings which I will call First Order Real Arithmetic (FORA), and Measure Arithmetic (MA). Now, FORA is similar to addition, multiplication, division, what have you, on the reals. However, we can only use finite formulas, so infinite sums cannot really be handled by FORA. FORA is the model theory perspective of real arithmetic and in the same way that natural arithmetic can have super big natural numbers, FORA can have super huge large real numbers, as well as super tiny real numbers (known as infinitesimals. 
Measure Arithmetic of the real line is where integration takes place. Practically speaking, integration is the way one sums real numbers in the way discrete sums don't really work. The problem is, suppose we want to compute something like $\sum_{i\in [0,1]} 1$. If we simply use integration, or in it's natural notation, $\int_0^1 1dx$, we compute that this sum is equal to  1.
Finally, as Jack D'Aurizio pointed out, infinite sums are defined as the limit of finite sums. There is more to be said about how nice of an infinite cardinal $\aleph_0$ is (practically, since it is the limit of finite cardinals (and we know how to manipulate finite cardinals), it gives us a lot of control over what can happen as things tend to it). But I hope this clears some concepts up!
Edit: Now that I think about it more, even in Logics like $L_{\mu,\kappa}$ you cannot discuss infinite sums over the reals. The major issue is that there is no standard method for dealing with terms of uncountable length. You can, however, discuss formulas with uncountable length. 
Addendum: I have just found an old paper by Keisler which deals with a formal system with infinitary functions. Check out section 5 if you are interested. But I should caution that it is pretty notationally dense and difficult to parse. 
