Generating Function for Sequence of Powers of Constant/Examples/2^n

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 $6$