Generating Function for Constant Sequence

Theorem

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N: a_n = r$

for some $r \in \R$.

Then the generating function for $\sequence {a_n}$ is given as:

$\map G z = \dfrac r {1 - z}$ for $\size z < 1$

$\ds \map G z$

$=$

$\ds \sum_{n \mathop = 0}^\infty r z^n$

$\ds $

$=$

$\ds r \sum_{n \mathop = 0}^\infty z^n$

$\ds $

$=$

$\ds \frac r {1 - z}$

for $\size z < 1$.

$\blacksquare$

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:

$\map G z = \dfrac {1 + z} {1 - z}$