There are several methods to say whether sum of series is finite or not. Can we say whether sum of series is countable or not.

For example $S_n=\Sigma_{0 \leq i \leq n}{2^i}= 2^{n+1}-1$ So for $n=\aleph_0$

$ S_{\aleph_0}=2^{\aleph_0}-1$

So can we say that S has a value which is not countable. But wouldn't that mean that sum of integers turns out be a non integer, as all integers form a set of countables. Or can we say that the series $S_i$ has a limit which lies outside set of inetgers so $\{S_1, S_2, S_3 ...\} \subset I$ has a limit $p$, such that $p \notin I$

Edit: I understood that $\{S_1, S_2, S_3 ...\} \subset I$ has a limit $p$ is incorrect as neither $\aleph_0$ nor $2^{\aleph_0}$ lie in $\mathbf{R}$ but I am still confused about other things.

Edit: Suppose we have sets $A_1, A_2, ... A_i...$ each having $2^i$ elements for $i=1,2 ...$ i.e. $\#(A_i)=2^i$. Suppose all these sets are disjoint can we say whether the set $A=\bigcup_{1 \leq i \leq \infty} A_i$ is countable or not.