Generating Function for Sequence of Powers of Constant/Examples
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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:
- $\map G z = \dfrac z {\paren {1 - b z} \paren {1 - b z - z} }$