Generating Function for Sequence of Harmonic Numbers

Theorem
Let $\left \langle {a_n}\right \rangle$ be the sequence defined as:
 * $\forall n \in \N_{> 0}: a_n = H_n$

where $H_n$ denotes the $n$th harmonic number.

That is:
 * $\left \langle {a_n}\right \rangle = 1, 1 + \dfrac 1 2, 1 + \dfrac 1 2 + \dfrac 1 3, \ldots$

Then the generating function for $\left \langle {a_n}\right \rangle$ is given as:
 * $G \left({z}\right) = \dfrac 1 {1 - z} \ln \left({\dfrac 1 {1 - z} }\right)$

Proof
Take the sequence:


 * $S_n = 1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots$

From Generating Function for Sequence of Reciprocals of Natural Numbers, this has the generating function:


 * $H \left({z}\right) = \ln \left({\dfrac 1 {1 - z} }\right)$

By definition, $\left \langle {a_n}\right \rangle$ is the sequence of partial sums of $\left \langle {a_n}\right \rangle$.

The result follows from Generating Function for Sequence of Partial Sums of Series.