Explanation of Z transform table

Question

I want to ask what is the Z transformation of n? How can be n, transformated into z/(z-1)^2?

Thank you

Answer 1

For $|z| < 1$ $$f(z) = \sum_{n=0}^\infty z^{n} = \frac{1}{1-z}, \qquad f'(z) = \frac{1}{(1-z)^2} = \sum_{n=0}^\infty n z^{n-1}$$ Thus for $|z^{-1}| < 1$ $$\mathcal{Z}[n 1_{n \ge 0}] = \sum_{n=0}^\infty n z^{-n} = z^{-1}f'(z^{-1}) = \frac{z^{-1}}{(1-z^{-1})^2} = \frac{z}{(z-1)^2}$$