Integral Expression of Harmonic Number

Theorem
Let $\sequence {H_n}_{n \in \N}$ 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
For $x \in \R _{\ge 1}$ let:
 * $\ds \map \phi x := \dfrac 1 x$

and:
 * $\ds \floor x = \sum _{1 \le k \le x} 1$

be the floor function.

Then: