Let $C_1,C_2,...$ be points on the same line as the $B's$, such that $C_1=B_1$ and $C_iC_{i+1}=1$. Then the infinite sum is just $\angle A_1C_1A_2+\angle A_1C_1A_2+...$. Let $D_i$ be the intersection of $A_2C_1$ and $A_1C_i$. Then it is easy to show that $\angle A_1C_1A_2+...+\angle A_1C_iA_2=\angle A_1D_iA_2$, since $A_1A_2C_{i+1}C_i$ is a parallelogram. Then as $i\to \infty$, $\angle A_1D_iA_2=180-\alpha$.