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 $$\left|{z}\right| < 1$$