A bogus proof of countable power set [duplicate]
The range of the function includes only the finite subsets of $\Bbb N$. If $S$ is an infinite subset of $\Bbb N$, then $\sum_{i\in S}2^i$ is not defined.
The mistake is that you assume that you assumed $S \in 2^{\Bbb N}$ is a finite set. In fact, your statement constitutes a valid proof that there are countably many finite subsets of $\Bbb N$, which is certainly a useful bit of information.
Note that if you look at infinite sums of the form $\sum_{i} k_i2^{-i+1}$, you end up with a surjective map to $[0,1]$.