Alternating harmonic series containing floor function

Let $$S\left(a\right)=\sum_{n=0}^\infty \frac{\left(-1\right)^{\left[na\right]}}{\left[na\right]+1}$$
where $a\gt 0$, and $\left[\cdot\right]$ denotes the floor function.
Consider $a\in \mathbb{Q}$, we can conclude that not all $a$ can make $S\left(a\right)$ converge.
In particular, we have $S\left(\frac{1}{k}\right)=k\ln 2$.
Let $$A=\left\{a\mid S\left(a\right)\:converges\right\}$$
My questions are:
•Does $A$ contain any irrational numbers?
•Is $A$ an uncountable set? A full-measure set?
•If so, I would guess the following limit $$\lim_{a\in A, a\rightarrow 0^+}a\, S\left(a\right)$$ exists and gets $\ln 2$. Am I right?