Generating Function for Sequence of Powers of Constant/Examples/2^n
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Example of Generating Function for Sequence of Powers of Constant
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:
- $\map G z = \dfrac 1 {1 - 2 z}$
Proof
A specific instance of Generating Function for Sequence of Powers of Constant:
- $\map G z = \dfrac 1 {1 - 2 z}$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-4}$ Generating Functions: Exercise $6$