Proof of Z-transform of n

Question

How can I prove the following Z-transform:

$$ Z\{n\} = \frac {z} {(z-1)^2} $$

As a tip, I was told to use the 'Multiplication in time'-property of the Z transform, which is the following:

$$ Z\{nx[n]\} = -z \frac{dX(z)}{dz} $$

But I don't see how I can use this.

Answer 1

Use the definition of $z$ transform

$$Z\left\{n\right\} = \sum_{n=0}nz^{-n}=\frac{z} {(z-1)^2}. $$

now you can use the geometric series (see my answer) to find the desired result.

Note: The $z$ transform of a sequence $a_n$ is given by

$$ \sum_{n=0}^{\infty} a_n z^{-n}.$$