$z$-transform of $1/n$

Question

How can one calculate the $z$-transform of: $x(n) = \frac{1}{n}$ , where $n \geq 1$? I have searched for table entries, then got stuck while trying to do it with the definition of $z$-transform (summation).

Answer 1

From what I can gather, by definition, the $z$-transform is the sum $$ \sum_{n = 1}^\infty\frac{z^{-n}}{n} $$ which just so happens to be the power series of $-\ln(1 - 1/z)$, converging as long as $|z| > 1$.