Question

Which of the following sequence \( f(n) \) has z-transform \( X(z) = e^{1/z} \)?

• A. \( \frac{1}{n!} u(n) \)
• B. \( -\frac{1}{n!} u(-n) \)
• C. \( (-1)^n \frac{1}{n!} u(n) \)
• D. \( \frac{u(-n-1)}{(n+1)!} \)

Hint:

The z-transform of \( \frac{1}{n!} u(n) \) is the exponential function \( e^{1/z} \).

Answer 1

The Correct Option is A

Z-transform of \( x(n) = \frac{1}{n!} u(n) \) is: \[ X(z) = \sum_{n=0}^{\infty} \frac{1}{n!} z^{-n} = e^{1/z} \]