Generating Function for Sequence of Reciprocals of Natural Numbers

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

That is:
 * $\left \langle {a_n}\right \rangle = 1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots$

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

Proof
Take the sequence:


 * $S_n = 1, 1, 1, \ldots$

From Generating Function for Constant Sequence, this has the generating function:


 * $\displaystyle G \left({z}\right) = \sum_{n \mathop = 1}^\infty z^n = \frac 1 {1 - z}$

By Integral of Generating Function:

which is the power series whose coefficients are $\left \langle {a_n}\right \rangle$.

But:
 * $G \left({z}\right) = \dfrac 1 {1 - z}$

and so by Primitive of Reciprocal and the Integration by Substitution:

The result follows from the definition of generating function.