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
Recall :

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

Let $N \ge n \ge 1$.

By Sum of Integrals on Adjacent Intervals for Integrable Functions:
 * $\ds \paren {H_n - \ln n} - \paren {H_N - \ln N} = \int_n^N \dfrac {u - \floor u} {u^2} \rd u$

We claim:
 * $\ds 0 \le \paren {H_n - \ln n} - \paren {H_N - \ln N} \le \frac 1 n$

Indeed, from follows:
 * $\forall u \in \R_{\ge 1} : 0 \le \dfrac {u - \floor u} {u^2} \le \dfrac 1 {u^2}$

In view of Integral Operator is Positive, integrating the above inequality on $\closedint n N$:

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

Thus the limit $\gamma$, the Euler-Mascheroni constant, exists by Cauchy's Convergence Criterion.

For each $n \in \N$, let $N \to \infty$.

Then:
 * $\ds \forall n \in \N : 0 \le H_n - \ln n - \gamma \le \dfrac 1 n$