The region of convergence are the values of *z* for which the *z*-transform converges. Equation (6.1) shows that *z*-transform is an infinite power series which is not always convergent for all values of *z*. Therefore, the region of convergence should be mentioned along with the *z*-transformation.

**Example 6.1** Find the *z*-transform of the following sequences

(i) *x*_{1}(*n*) = {1, 2, 3, 4, 5, 6, 7} and (ii) *x*_{2}(*n*) = {1, 2, 3, 4, 5, 6, 7}

↑

**Solution**

- For the given sequence
*x*_{1}(*n*) = {1, 2, 3, 4, 5, 6, 7}, we can write*x*_{1}(0) = 1,*x*_{1}(1) = 2,*x*_{1}(2) = 3,*x*_{1}(3) = 4,*x*_{1}(4) = 5,*x*_{1}(5) = 6 and*x*_{1}(6) = 7.The

*z*-transform of the sequence*x*_{1}(*n*) is given byIn the present case

*n*= 0 to*n*= 6.*X*_{1}(*z*) has finite ...

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