\[|X(z)|=\left|\sum_{n=-\infty}^{\infty} x[n] z^{-n}\right| \leq \sum_{n=-\infty}^{\infty}\left|x[n] z^{-n}\right|=\sum_{n=-\infty}^{\infty}|x[n]|(|z|)^{-n} \nonumber \] \[N(z) \leq C_{1} \sum_{n=-\in...\[|X(z)|=\left|\sum_{n=-\infty}^{\infty} x[n] z^{-n}\right| \leq \sum_{n=-\infty}^{\infty}\left|x[n] z^{-n}\right|=\sum_{n=-\infty}^{\infty}|x[n]|(|z|)^{-n} \nonumber \] \[N(z) \leq C_{1} \sum_{n=-\infty}^{-1} r_{1}^{n}(|z|)^{-n}=C_{1} \sum_{n=-\infty}^{-1}\left(\frac{r_{1}}{|z|}\right)^{n}=C_{1} \sum_{k=1}^{\infty}\left(\frac{|z|}{r_{1}}\right)^{k} \nonumber \] The Region of Convergence is the area in the pole/zero plot of the transfer function in which the function exists.
