TITLE: Prove that the sequence converges absolutely
QUESTION [6 upvotes]: Let $\{a_n\}$ and $\{r_n\}$ be two sequences of real numbers such that $\sum_{n=1}^\infty |a_n|< \infty.$ Prove that
$$
\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{\sqrt{|x-r_n|}}
$$
converges absolutely for almost every $x \in \mathbb{R}$.
Can anyone provide a useful hint to solve the problem ? I am unable to figure out how does almost every $x$ come into picture. Should I use some lebesgue integral ?

REPLY [4 votes]: For any bounded measurable $A$
$$\int_A\sum_{n=1}^{\infty}\frac{|a_n|}{\sqrt{|x-r_n|}}=\sum_{n\ge 1}\int_A\frac {|a_n|}{\sqrt{|x-r_n|}}<\infty$$
Hence the sum is finite a.s. This can be extended to show a.s. absolute convergence of $$\sum_{n=1}^{\infty}a_nf_n(x)$$
for any locally uniformly integrable $f_n$.