The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is
- $1 /\left(z^{-1}-a\right)$, region of convergence $\text{(ROC)}$: $|z| \leq 1 / a$
- $1 /\left(1-a z^{-1}\right)$, $\text{ROC}$: $|z| \geq a$
- $1 /\left(1-a z^{-1}\right)$, $\text{ROC}$: $|z|>a$
- Item $(a)$ if $a>1$, Item $(b)$ if $a<1$
- $\exp \left(a z^{-1}\right)$, $\text{ROC}$: entire complex plane.