Integral Expression of Harmonic Number

Theorem
Let $\sequence {H_n}_{n \mathop \in \N}$ be the harmonic numbers.

Then:
 * $\ds H_n = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u$

where $\floor u$ denotes the floor of $u$.

Proof
Observe that:
 * $\ds \forall x \in \R_{\ge 1} : \floor x = \sum_{1 \mathop \le k \mathop \le x} 1$

Let $\phi : \R_{\ge 1} \to \R$ be defined as:
 * $\ds \map \phi x := \dfrac 1 x$

Then: