Generating Function for Sequence of Powers of Constant/Examples

Examples of Generating Function for Sequence of Powers of Constant

Example: $\sequence {2^n}$

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

$\forall n \in \Z_{\ge 0}: a_n = 2^n$

That is:

$\sequence {a_n} = 1, 2, 4, 8, \ldots$

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

$\map G z = \dfrac 1 {1 - 2 z}$

Example: $\sequence {2^n + 3^n}$

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

$\forall n \in \Z_{\ge 0}: a_n = 2^n + 3^n$

That is:

$\sequence {a_n} = 2, 5, 13, 35, \ldots$

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

$\map G z = \dfrac 1 {1 - 2 z} + \dfrac 1 {1 - 3 z}$

Example: $\sequence {\paren {b + 1}^n - b^n}$

Let $b \in \R_{>0}$ be a positive real number.

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

$\forall n \in \Z_{\ge 0}: a_n = \paren {b + 1}^n - b^n$

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

$\map G z = \dfrac z {\paren {1 - b z} \paren {1 - b z - z} }$