Derivative of 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.

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

from Generating Function for Sequence of Harmonic Numbers.

Then the derivative of $G \left({z}\right)$ $z$ is given by:


 * $G' \left({z}\right) = \dfrac 1 {\left({1 - z}\right)^2} \ln \left({\dfrac 1 {1 - z} }\right) + \dfrac 1 {\left({1 - z}\right)^2}$