I tried two approaches and gained the different conclusions of judging the stability of the transfer function of the system
The system is stable. Your mistake is in the application of the formula for the geometric series:
$$\begin{align}-\frac{3}{2+z^{-1}}&amp;=-\frac32\frac{1}{1+\frac12z^{-1}}\\&amp;=-\frac32\sum_{n=0}^{\infty}\left(-\frac12\right)^nz^{-n}\end{align}$$
Hence, the system's impulse response is
$$h[n]=2\delta[n]-\frac32\left(-\frac12\right)^nu[n]$$