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:

$\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$

Take the sequence:

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

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

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

$\blacksquare$