Question on cardinality of a set

Does anyone know the size of this set or how to figure it out:

$$ left{0.a_1a_2a_3dots colon sum_{i=2}^{infty}frac{1}{G_i}inmathbb{R}right},$$
where $G_i=(9-a_{(sum_{j=1}^{i-1}j)+1})dots(9-a_{(sum_{j=1}^{i-1}j)+i})$ and if $G_i=0$, define $frac{1}{G_i} = 0$.

e.g. for $ 0.dot{1}21314151617181dot{9}dots$, $G_2=(9-2)(9-1)=78$, $G_3=(9-3)(9-1)(9-4)=685$, $G_4=(9-1)(9-5)(9-1)(9-6)=8483$ and so on.