Let $\Omega_r =\{(x,y,z) \in \mathbb{R}^3 : 0<x,y,z , x^2+y^2+z^2<r^2 , x^2+y^2-z^2<0 \}$
And $ f_r(x,y,z)= \begin{cases} 2z/(x^2+y^2+z^2) & \text{if } (x,y,z) \in \Omega_r, \\ 0 & \text{otherwise}. \end{cases}$
Show that the serie $\sum 2^{-n}f_n $ converges a.e. in $ \mathbb{R}^3 $ to an integrable function in $\mathbb{R}^3$
I have tried applying quotient criterion and it gives me $1/2 < 1$ so I can think that it is convergent. I have tried to find another function which is integrable and bounds my function but so for example if $ x, y$ or $z \ge1 $ then $f_n\le 1/2^{n-1} $, which is convergent and integrable and if $ z < 1 $ then $f_n< 1 /(2^{n-1}(x^2+y^2+z^2 ) $ but I don't know if that is enough.
I need help. Thank you in advance.