Euler's Integral Theorem

Theorem

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

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

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

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

In particular, $\sequence {H_n - \ln n}$ is a bounded decreasing sequence.

Thus the limit $\gamma$ exists by monotone convergence theorem.

Let $N \to \infty$.

By definition of the Euler-Mascheroni constant:
 * $\ds 0 \le H_n - \ln n - \gamma \le \dfrac 1 n$