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

$\blacksquare$