I was wondering if it was possible to express infinite divergent sums using set theory (I do apologize if some of the notation is improper)
I wanted to consider $$\sum_{n=1}^{\infty}n$$ i.e. the sum of all natural numbers. I thought you might be able to express this by adding up the cardinality of sets with n elements. So
$card(\{1\}) +card(\{2,3\})+card(\{4,5,6\})+...$ and so on. This then would equal
$card(\{1\} \bigcup\{2,3\}\bigcup\{4,5,6\}\bigcup...)=\mathbb{N}$
Hence $$\sum_{n=1}^{\infty}n=\aleph_0$$
Is my math correct and can this be generalized to compute other infinite sums, even those which involve terms with decimals?
Note that using your notation, $$ \sum_{n=1}^\infty 1 = \aleph_0 $$ and the LHS sum is quite simpler than yours. In fact, I do not understand how you can get anything more than $\aleph_0$ on the RHS at all...