In general can we say $\infty=\infty$? Eg, $\sum_{i\in \mathbb{N}} i =\sum_{i \in \mathbb{Q}_+}i$ This might be a bit of a basic question but my current understanding is that we cannot. Still it makes me wonder if we can propose a mapping between two countable sets why not?
For example why is this expression incorrect?
$$\sum_{i\in \mathbb{N}} i =\sum_{i \in \mathbb{Q}_+}i$$
Since our LHS and RHS are both positive infinity.
Any help is appreciated
 A: I am assuming that, for you $\Bbb Z^+$ is $\{0,1,2,3,\ldots\}$. If that so, no, you don't have$$\sum_{i\in\Bbb N}i=\sum_{i\in\Bbb Z^+}i.\tag1$$That's so because the LHS of $(1)$ is the series$$0+1+2+3+4+\cdots,$$whereas the RHS of $(1)$ is the series$$1+2+3+4+5+\cdots$$These are two distinct series.
On the other hand, you do have$$\sum_{i=0}^\infty 2^{-i}=\sum_{i=1}^\infty i2^{-i},\tag2$$although, again you have two distinct series. But this time you have two convergent series and the sum of both sides of $(2)$ is $2$. It happens that the notation $\sum_{i\geqslant k}a_i$ is ambiguous; it is both a series and its sum (when the series converges). But there is not ambiguity in the case of $(1)$, since non of the series converges.
Note that it is not correct that both sides of $(1)$ are equal to $\infty$. What happens is that both series diverge to $\infty$.
A: First of all, let's get the stuff about the mapping between countable sets out.
The fact that we can construct a bijective mapping between countable sets means exactly that, we can send every element of one set to one and only one in the other.
Note that this doesn't say that the mapping has anything to do with how it does it, it just means it's possible to do.
For this reason, knowing that $\mathbb{Q}$ and $\mathbb{N}$ are in bijection tells us nothing about those series. In fact, any series is indexed by a countable set, but not all series diverge to $\infty$ !
Now to the meat of the question. I assume that you mean that both those series diverge to infinity. First of all, you should be very careful on saying what you mean by   $$\sum_{i \in \mathbb{Q}_+}i$$
But rather than trying to make sense of it, which is not actually terribly important to the arguement, substitute it with any divergent series to $+\infty$ , let's say $$\sum_{n=0}^{\infty}a_n$$
Now you ask, is this expression true?
$$\sum_{n=0}^{\infty}a_n=\sum_{i\in\Bbb N}i$$
The answer would be in fact yes, but not in the way you imagine it. We can consider the extended real line $\mathbb{R}\cup\{\pm\infty\}$ and in fact both of those values are the same, i.e. $+\infty$ . Great! But wait a minute, now we should be able to subtract $-\infty$ from both sides and get $+\infty-\infty=0$, right? No.
The reason why this can't be done is that, any new set you define is just a set, and then you can start to define operations on this new set. For subsets of the real numbers, you can just use the operations on $\mathbb{R}$ , but this set is bigger, so we need to define a new operation so that we can say what it means to sum a number and $+\infty$, or multiply and add $+\infty$ and $-\infty$ . The sad truth, though, is that we can't really do the operations in a way that perfectly extends our operations on the reals, makes it so that every element has a multiplicative and additive inverse AND makes the infinities behave like you would expect them to.
What all this means is that, yes, you can construct a system where divergent sequences(series) are convergent to $+\infty$ (using an appropriate metric space structure), but you can't also have a well defined operation that behaves like you want, so while we can say that yes, in fact, $+\infty=+\infty$ and $-\infty=-\infty$, we can't say that $\infty-\infty=0$ and $-{\infty}+{\infty}=0$ . The first expression simply means that they are equal as elements of the set, while the second would imply an operation that we haven't defined.
