Let us have the set S defined as follows:
S:= { $X\subseteq \mathbb N: \sum_{x\in X} x=\infty $}
thus S is the set of all subsets X of N such that the sum of the numbers in each set X is infinite.
I tend to say that S is countable even though I am not sure. We know that the subsets of a countable set are countable and since N is countable then all X will be countable. Since the sum of any X is infinite then any subset X will be countably infinite (?) This means we have the set of all countably infinite subsets of N. Let S be {$x_{1}$,$x_{2}$,$x_{3}$...} where $x_{i}$ is a countably infinite set. We can define a function $f: S \to \mathbb N$ such that f($x_{i}$)=i. Since f is a bijection to N then S is countable?
I'm really unsure on the last part (how to define the bijective function). Anybody that could point me to the right direction and tell me if my logic so far is alright?
$S$ has the cardinality of continuum since it is $\mathcal P(\mathbb N) - \mathcal P_f (\mathbb N)$, where $\mathcal P_f(\mathbb N)$ is the set of finite subsets of $\mathbb N$, which is countable (easy exercise).