What is the sum of all positive rational numbers? Can it be defined? What type of infinity is it?
The question is, quite simply, incredibly ill-defined.
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Infinite Limits:
When we say a summation approaches or diverges to infinity, this is not the "infinity" of the aleph numbers that you're thinking of. The infinity in this context is more of a notion or concept, rather than an actual number itself.
To see what I mean, recall: an infinite summation can be defined as the limit of its partial sums, i.e. for a sequence $(a_n)_{n \in \Bbb N}$,
$$\sum_{n=0}^\infty a_n \eqdef \lim_{N \to \infty} \sum_{n=0}^N a_n$$
Now, to further this discussion, we have to appeal to the formal, rigorous definition of a sequence approaching infinity as its index approaches infinity: a sequence $(s_n)_{n \in \N}$ is said to approach infinity if
$$(\forall R \in \R)(\exists N \in \N)(\forall n \ge N)(s_n > R)$$
In words: $s_n \to \infty$ if and only if...
- ...for every real number $R$...
- ...there exists some natural number $N$ such that...
- ...for every integer $n$ greater than or equal to $N$...
- ...it holds that $s_n > R$
(Of course, $s_n$ in our case is your sequence of partial sums.)
This is quite a mouthful, but essentially it says that, for any real number, the sequence is eventually always greater than that real number. It encodes the notion and intuition of divergence to infinity.
Notice in particular: this definition does not, in any way, actually reference infinity itself. This is what I mean when I say this infinity is more conceptual than an actual thing - it is not inherently a number itself.
Infinite Cardinals:
What about the infinities that, say, describe how many numbers there are in a set? You have presumably seen, for instance, we say that
- the cardinality of the naturals, denoted $|\N|$, is equal to $|\aleph_0|$
- moreover, $|\Bbb Q| = |\Bbb Z| = \aleph_0$
- however, $|\R| = |\Bbb C| = |(0,1)| = c$ (or $\aleph_1$, depending on the continuum hypothesis)
- the set $\R^\R$ of all functions $f : \R \to \R$ has cardinality $2^c$
...and so on. But what does this mean?
We call these "cardinal numbers." Particularly, these are what we call "infinite cardinals." Cardinality is a means of measuring how many elements are in a set. For a finite set, it is easy enough:
$$|\set{0,1,2}| = 3 \qquad |\set{1}| = 1 \qquad |\set{1,2,\cdots,100}| = 100 \qquad |\varnothing|=0$$
and so on. To generalize things to infinite sets, however, makes the definition a bit murkier. Assuming the axiom of choice, we say that the cardinality of a set $S$ is given by the least ordinal number $\alpha$ such that there exists a bijective (i.e. invertible) function $f : S \to \alpha$. Ordinal numbers are a bit more complicated to get into; the two are intuitively similar but formally not at all the same, albeit intimately linked.
That said, essentially, we then can say that two sets have equal cardinality if there exists a bijection between them - and this works for infinite sets as well as finite sets. Moreover, we can say $|S| \le |T|$ if there is an injective function $f : S \to T$, and $|S| \ge |T|$ if there is a surjective function $g: T \to S$.
Uncountable Sums:
The main takeaway from this is that the infinities used to describe cardinality are precisely that: they are used to describe how many elements a set has. On the other hand, the infinity from limits is a purely conceptual construct, and is not inherently representative of any such infinity.
However, there is slight detour I want to go on: sums on arbitrary sets.
In a loose sense, you can imagine the usual infinite sum as being a sum over $\N$ or another countable set. (A set is countable if its cardinality is at most $\aleph_0$.) Perhaps we can generalize the concept? You can this talked about here on MSE, but essentially the takeaway is this: we can sum over any set $S$ whose members are nonnegative by
$$\sum_{i \in S} a_i \eqdef \sup_{F \subseteq S} \set{ \sum_{i \in F} a_i \; \middle| \; F \subseteq S \text{ is finite} }$$
Ultimately, then, what this means is that the sum over a set is the least upper bound on the value of any finite sum we can get. However this ultimately means there are at most countably many nonzero summands $a_i$ and thus can be reduced to some finite or (typical) infinite sum - so it's not particularly useful or noteworthy in general. Figured it'd be neat to bring up though.
What is the sum of $\aleph_0$ numbers? Again, can it be defined? What does it depend on? What about $\aleph_i$ numbers?
If by this you mean "what is the sum of a set of $\aleph_i$-many numbers?", I guess the above explains it.
Of course, we can take the sums of these numbers, we just need at most countably many nonzero such numbers. And of course, by merit of how I started this discussion, the sum ultimately still has to be defined.
In the general case, I'm not sure if there is a known necessary and sufficient condition for an arbitrary infinite series $\sum a_n$ (of rationals or anything else) to exist. One of the main necessary conditions is that $a_n \to 0$, but it is not sufficient (consider $a_n = 1/n$). Some particular types of series do have convergence tests that you can look at, at least.
What is the sum of all positive rational numbers?
I suppose we can use the spirit of the definition I gave earlier.
Define the set $\N_n \eqdef \set{1,2,\cdots,n}$. Observe that $\N_n \subseteq \Q^+$ and it is finite. Thus, from the definition of the sum and of supremum,
\begin{align*}
\sum_{r \in \Q^+} r
&= \sup_{F \subseteq \Q^+} \set{ \sum_{r \in F} r \; \middle| \; F \subseteq \Q^+ \text{ is finite} }\\
&\ge \sum_{r \in \N_n} r
\end{align*}
and this holds for every $n \in \N$. Observe that, moreover,
$$\sum_{r \in \N_n} r < \sum_{r \in \N_{n+1}} r \qquad \& \qquad \sum_{r \in \N_n} r = \frac{n(n+1)}{2}$$
Therefore, taking the limit to infinity,
$$\sum_{r \in \Q^+} r \ge \lim_{n \to \infty} \sum_{r \in \N_n} r = \lim_{n \to \infty} \frac{n(n+1)}{2}$$
which is clearly infinite, and thus the sum of the positive rationals does not converge. (Again, we do not say "what kind" of infinity it is, simply because that question is not really well-defined, or is nonsensical really, per the earlier discussion.)
Can we tell anything about how big the sum of $\aleph_0$ or even $\aleph_i$ numbers can get?
Since every summation (within the definitions I've given) are limited to essentially countable sums at most, bear that in mind going forward.
As you noted, we can essentially make a sequence of rational numbers approach any real number. (This is because $\R$ forms what we call a "complete metric space" and in particular is what we call the "completion" of the metric space for $\Q$.) In particular, if you want a sequence of rational numbers that approach $r \in \R$, simply define
$$r_n \eqdef \frac{ \lfloor 10^n r \rfloor}{10^n}$$
If you want a summation that does so, take
$$s_n \eqdef \frac{ \lfloor 10^n r \rfloor - 10^n \sum_{i=0}^{n-1} s_i }{10^n} , s_0 = \lfloor r \rfloor$$
and then $\sum_{n=0}^\infty s_n = r$. (Essentially just a formal way of summing up the digits of the number, relative to their base-$10$ representation.)
Thus, within the real numbers, there isn't really a "limit" as to how big a sum can get: it can be any finite number you want. Of course, there is also infinity (which is often defined as being greater than every real number), but if you include that then $\infty$ is the absolute most you can achieve (because once again, there aren't different "kinds" of this particular conceptualization of infinity).
