Unilateral Z-transform

Remarks

Let $r = \limsup_{n \to \infty} \abs{x_n}^{1/n}$. The series converges absolutely for $\abs{z} > r$ and diverges for $0 < \abs{z} < r$, by the root test. Example: $x_n = n$ gives $X(z) = \sum_{n \ge 0} n z^{-n} = z/(z - 1)^2$ for $\abs{z} > 1$, by differentiating the geometric series. Non-example: $x_n = n!$ gives $r = \infty$, so the series converges for no $z$, and $x$ has no unilateral Z-transform as a function. The unilateral transform of $x$ equals the Z-transform of the sequence that agrees with $x$ for $n \ge 0$ and is $0$ for $n < 0$. Writing $w = z^{-1}$ turns it into the ordinary generating function $\sum_{n \ge 0} x_n w^n$.