Generating Function for Sequence of Harmonic Numbers

Theorem
Let $\sequence {a_n}$ 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:
 * $\sequence {a_n} = 1, 1 + \dfrac 1 2, 1 + \dfrac 1 2 + \dfrac 1 3, \ldots$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$

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:


 * $\map H z = \map \ln {\dfrac 1 {1 - z} }$

By definition, $\sequence {a_n}$ is the sequence of partial sums of $\sequence {a_n}$.

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