Question

The value of the radius of convergence of f(n) = 2^n, n<0 is ____________

(a) 0<| z |<1

(b) -2<| z |

(c) | z |<2

(d) z-plane

I had been asked this question in class test.

The doubt is from Region of Convergence in division Laplace Transform and System Design of Signals and Systems

Answer 1

The correct option is (c) | z |<2

For explanation I would say: Z [f (n)] = \(∑_{-∞}^∞ f(n) z^{-n}\)

 = \(\frac{z}{2} + \frac{z^2}{2} + \frac{z^3}{2} + …. ∞\)

= \(\frac{z}{2}(1 + \frac{z}{2} + \frac{z^2}{2} + …..∞)\)

= \(\frac{1}{1-\frac{z}{2}} . \frac{z}{2} = \frac{z}{2-z}\)

The series is convergent if | \(\frac{z}{2}\) |<1, that is | z |<2

Hence, ROC is | z |<2.