Generating Function for Sequence of Reciprocals of Natural Numbers

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

That is:
 * $\sequence {a_n} = 1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots$

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

Proof
Take the sequence:


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

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


 * $\displaystyle \map G z = \sum_{n \mathop = 1}^\infty z^n = \frac 1 {1 - z}$

By Integral of Generating Function:

which is the power series whose coefficients are $\sequence {a_n}$.

But:
 * $\map G z = \dfrac 1 {1 - z}$

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

The result follows from the definition of generating function.