Generating Function for Constant Sequence/Examples/a0=1, an=2

Example of Generating Function for Constant Sequence
Let $\sequence {a_n}$ be the sequence defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = \begin{cases} 1 & : n = 0 \\ 2 & : n > 0 \end{cases}$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\map G z = \dfrac {1 + z} {1 - z}$

Proof
Let $\map {H_1} z$ be the generating function for $\sequence {r_n}$ where:
 * $r_n = 2$

Then from Generating Function for Constant Sequence:
 * $\map H z = \dfrac 2 {1 - z}$

Then: