Generating Function for Natural Numbers

Theorem
Let $\left \langle {a_n}\right \rangle$ be the sequence defined as:
 * $\forall n \in \N_{> 0}: a_n = n$

That is:
 * $\left \langle {a_n}\right \rangle = 1, 2, 3, 4, \ldots$

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $\displaystyle G \left({z}\right) = \frac 1 {\left({1 - z}\right)^2}$

Proof
Take the sequence:


 * $S_n = 1, 1, 1, \ldots$

From Generating Function for Constant Sequence, this has the generating function:


 * $\displaystyle G \left({z}\right) = \sum_{n \mathop = 1}^\infty z^n = \frac 1 {1 - z}$

By Derivative of Generating Function:


 * $\displaystyle \dfrac \d {\d z} G \left({z}\right) = 0 + 1 + 2 z + 3 z^2 \cdots = \sum_{n \mathop \ge 0} \left({n + 1}\right) z^n$

which is the generating function for the sequence $\left \langle {a_n}\right \rangle$.

But $G \left({z}\right) = \dfrac 1 {1 - z}$

and so by Power Rule for Derivatives and the Chain Rule:
 * $\dfrac \d {\d z} G \left({z}\right) = \dfrac 1 {\left({1 - z}\right)^2}$

The result follows from the definition of a generating function.