Harmonic Number as Unsigned Stirling Number of First Kind over Factorial

Theorem

 * $H_n = \dfrac { {n + 1} \brack 2} {n!}$

where:
 * $H_n$ denotes the $n$th harmonic number
 * $n!$ denotes the $n$th factorial
 * $\ds { {n + 1} \brack 2}$ denotes an unsigned Stirling number of the first kind.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $H_n = \dfrac { {n + 1} \brack 2} {n!}$

$\map P 0$ is the case:

Thus $\map P 0$ is seen to hold.

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $H_k = \dfrac { {k + 1} \brack 2} {k!}$

from which it is to be shown that:
 * $H_{k + 1} = \dfrac { {k + 2} \brack 2} {\paren {k + 1}!}$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 0}: H_n = \dfrac { {n + 1} \brack 2} {n!}$