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If x[n] = (1/3)^(|n|) - (1/2)ⁿ u[n], then the region of convergence (ROC) of its Z-transform in the z-plane will be

1. 1/3 < |z| < 3
2. 1/3 < |z| < 1/2
3. 1/2 < |z| < 3
4. 1/3 < |z|

Correct answer: 1/2 < |z| < 3

Solution

(1/3)^|n| is two-sided with ROC 1/3<|z|<3; (1/2)^n u[n] is right-sided with ROC |z|>1/2. The overall ROC is the intersection, 1/2<|z|<3, not 1/3<|z|.

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