Generating Function of Sequence by Index

Theorem
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Then:
 * $z G' \left({z}\right)$ is the generating function for the sequence $\left\langle{n a_n}\right\rangle$

where $G' \left({z}\right)$ is the derivative of $G \left({z}\right)$ $z$.

Proof
The result follows by definition of generating function.