What is the measure of the set of values $S=\{0.x_1x_2... \in (0,1)$ (in binary) for which $\sum_{n \geq 1}\frac{(-1)^{x_n}}{n}$ converges$\}$?

I think the measure is zero but I'm not sure. I'm pretty sure the set is measurable because its construction does not require Axiom of Choice. I don't know where to begin.

Edit: It was shown to be 1 by Kolmogorov's three series theorem.

A further question: Consider the function $f:(0,1)\rightarrow\mathbb{R}$ mapping $x$ values to the value of the corresponding series, and to $0$ otherwise. This function maps every open set to the whole real line. Does it map every set of positive measure to the whole real line?