Discrete-time Convolution Convergence Issue
In the third edition of Oppenheim and Schafer's Discrete-Time Signal Processing, the corresponding exercise is given with $h[n]=a^nu[-n-1]$, $a&gt;1$ (note the positive sign in the power of $a$). In that case everything works out fine. So either there's a typo in your edition or you made a mistake copying the exercise.