Consider $\{r_{n}\}_{n \ge 1}$ set of all rational numbers of $[0,1]$.
Now lets define $$\displaystyle S_{k}(x) := \sum_{n=1}^{k} \frac{1}{n^{2} \sqrt{|x-r_{n}|}}$$
My question is : does $S_k(x)$ converge almost everywhere on $[0,1]$\ $\mathbb{Q}$?
I thought about considering $\displaystyle \lim_{k \to \infty} \int_{[0,1]} S_{k}(x)\, dx$ and use Lebesgue theorem about limit under integration. Then if exact result is converges , I could say if my series converges a.e. or not. Am I right?
However , can we say something about this series without Lebesgue integral?
Since the $S_k(x)$ is pointwise monotone-inscreasing, we may use the montone convergence theorem to conclude that $$ \int_0^1 \lim_{n \rightarrow \infty} S_n(x) \mathop{dx} = \lim_{n \rightarrow \infty} \sum_{k=1}^n \frac{1}{k^2} \int_0^1 \frac{1}{\sqrt{|x-r_n|}} \mathop{dx} \leq 4 \sum_{k=1}^\infty \frac{1}{k^2} <\infty.$$ Thus, $S(x) := \lim_{n \rightarrow \infty} S_n(x) < \infty$ for $\lambda$-almost all $x \in [0,1]$.