Generating Function for Constant Sequence

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

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

Proof
Follows directly from Sum of Infinite Geometric Progression:


 * $\displaystyle G \left({z}\right) = \sum_{n=0}^\infty r z^n = r \sum_{n=0}^\infty z^n = \frac r {1-z}$ for $\left|{z}\right| < 1$