Convergence range of the spectrum of stationary process

Question

I'm working on a question regarding the convergence range of the spectrum of a discrete-time stationary process, which is defined as $$\phi(z)\triangleq\sum_{n=-\infty}^{+\infty}\gamma(\tau)z^{-n}$$ where $\gamma(\tau)$ is the covariance function. The author points out directly that the convergence range is a strip $r_1<|z|<r_2$ where $0<r_1<1$ and $r_2>1$. The spectrum function is apparently a Laurent series. I use d'Alembert's ratio test and find that the author's claim indicates that $$|\lim_{n=+\infty}\frac{\gamma(n+1)}{\gamma(n)}|<1 \qquad (1)$$ If the stationary process is Markovian Gaussian stationary process, then it decays exponentially. Other than that, is there a general proof of the convergence range of the spectrum of a stationary process? In other words, why does Eq. (1) always hold for stationary process. Thanks in advance for your help!