Euler's Integral Theorem

Theorem

 * $\ds H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$

where:
 * $H_n$ denotes the $n$th harmonic number
 * $\gamma$ denotes the Euler-Mascheroni constant.

Proof
For all $n \in \N_{>0}$:

Thus for all $N \ge n \ge 1$:

Letting $N \to \infty$ by :
 * $\ds 0 \le H_n- \ln n - \gamma \le \dfrac 1 n$