TITLE: The set of all binary sequences $(a_n)$ such that $\sum_{n=1}^{\infty} \frac{(-1)^{a_n}}{n}$ converges/diverges.
QUESTION [5 upvotes]: I am interested in knowing whether both the sets described above are uncountable. 
I think I managed to show that the set of binary sequences such that the series converges is uncountable. The main idea was to pick an arbitrary real $x$. Then given that $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, we can always construct $a_n$ such that the partial sums get arbitrarily close to $x$ and hence converge to $x$. Thus there is a surjection from all such sequences to the reals, so the set is uncountable.
I have had no luck with the set of sequences such that the series doesn't converge though. I tried the diagonal argument, but I cannot see a reason why the 'new sequence' would necessarily yield a divergent series. Any ideas?

REPLY [2 votes]: The set of divergent series is also uncountable. Choose $a_{2k}=1$, so that the sub-sequence of even terms already tends to infinity. Now, like in your part of the proof, let the odd terms tend to a real numbers $x$. The combination still diverges to infinity, so we have found uncountably many divergent series.