The z-transform of a discrete-time sequence $x[n]$ is defined as:

$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$

Calculating the Z-Transform ROC

Given the sequence:

$$x[n] = \begin{cases} (0.2)^n, & 0 \le n \le 7 \\ 0, & \text{otherwise} \end{cases}$$

The z-transform $X(z)$ is:

$$X(z) = \sum_{n=0}^{7} (0.2)^n z^{-n}$$

This can be rewritten as:

$$X(z) = \sum_{n=0}^{7} (0.2 z^{-1})^n$$

Geometric Series Convergence

This represents a finite geometric series. The region of convergence (ROC) is the set of z values for which the z-transform sum converges.

The terms in the summation are $(0.2)^n z^{-n}$. For these terms to be defined, $z^{-n}$ must be defined. The term $z^{-n}$ is undefined when $z=0$ for any $n > 0$. Since the sum includes terms for $n=1$ through $n=7$, the value $z=0$ must be excluded from the ROC.

The sum can be explicitly written as:

$$X(z) = 1 + (0.2)z^{-1} + (0.2)^2 z^{-2} + \dots + (0.2)^7 z^{-7}$$

This finite sum converges for all values of $z$ except $z=0$. The term $z=\infty$ does not cause divergence because $z^{-n} \to 0$ for $n>0$ as $z \to \infty$. The specific value $z=0.2$ results in a finite sum: $\sum_{n=0}^{7} (0.2)^n (0.2)^{-n} = \sum_{n=0}^{7} 1 = 8$. Thus, $z=0.2$ is included in the ROC.

Region of Convergence Determination

The ROC is determined by the values of $z$ for which the sum $X(z)$ exists. Based on the analysis of the $z^{-n}$ terms, the ROC excludes $z=0$. Therefore, the ROC consists of all values of $z$ except $z = 0$.