It works sometimes

$\large \lim_{n\to\infty} \sum_{s=1}^n \sum_{r=1}^n \bigg [ A(rs) \times 2^r 3^{-s} \bigg ]$

Consider the function $A(rs) = \begin{cases} {0 \quad , \quad r \ne s } \\ {1 \quad , \quad r=s } \end{cases}$ .

Evaluate the series above.