z-transform of two given sequences

Definition: F(z) = Σ f(k) z⁻ᵏ.

(i) f(k) = {15, 10, 7, 4, 1, −1, 3, 6}

The origin (k = 0) is at the fourth term (4), as read from the arrow. So the terms are:

k    : -3  -2  -1   0   1   2   3   4
f(k) : 15  10   7   4   1  -1   3   6

F(z) = 15 z³ + 10 z² + 7 z + 4 + z⁻¹ − z⁻² + 3 z⁻³ + 6 z⁻⁴

The sequence is of finite length and two-sided, so the ROC is the whole z-plane except z = 0 and z = ∞.

(ii) f(k) = 5ᵏ for k < 0; 2ᵏ for k ≥ 0

Split into two parts.

Part k < 0: Σ (k = −∞ to −1) 5ᵏ z⁻ᵏ = Σ (m = 1 to ∞) (z/5)ᵐ (with m = −k) = (z/5)/(1 − z/5) = z/(5 − z), converges for |z| < 5.

Part k ≥ 0: Σ (k = 0 to ∞) (2/z)ᵏ = 1/(1 − 2/z) = z/(z − 2), converges for |z| > 2.

Add:

F(z) = z/(5 − z) + z/(z − 2) = z [ (z − 2) + (5 − z) ] / [ (5 − z)(z − 2) ]

F(z) = 3z / [ (5 − z)(z − 2) ], ROC: 2 < |z| < 5