# 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
$\map \OO {\dfrac 1 n}$ denotes big-$\OO$ of $\dfrac 1 n$.

## Proof 1

Recall the definition of the floor function:

The floor function of $x$ is the unique integer $\floor x$ such that:

$\floor x \le x < \floor x + 1$

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

 $\ds H_n - \ln n$ $=$ $\ds 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u - \ln n$ Integral Expression of Harmonic Number $\ds$ $=$ $\ds 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u - \int_1 ^n \dfrac 1 u \rd u$ Definition of Real Natural Logarithm $\ds$ $=$ $\ds 1 - \int_1 ^n \dfrac {u - \floor u} {u^2} \rd u$ Integral Operator is Linear

Let $N \ge n \ge 1$.

$\ds (1): \quad \paren {H_n - \ln n} - \paren {H_N - \ln N} = \int_n^N \dfrac {u - \floor u} {u^2} \rd u$

On the other hand, from Definition of Floor Function 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 $\closedint n N$:

 $\text {(2)}: \quad$ $\ds 0$ $\le$ $\ds \int_n^N \dfrac {u - \floor u} {u^2} \rd u$ $\ds$ $\le$ $\ds \int_n^N \dfrac 1 {u^2} \rd u$ $\ds$ $=$ $\ds \dfrac 1 n - \dfrac 1 N$ $\ds$ $\le$ $\ds \dfrac 1 n$

From $(1)$ and $(2)$, it follows:

$(3): \quad 0 \le \paren {H_n - \ln n} - \paren {H_N - \ln N} \le \dfrac 1 n$

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.

In $(3)$, for each $n \in \N$, let $N \to \infty$.

Then:

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

$\blacksquare$

## Proof 2

Recall the definition of the floor function:

The floor function of $x$ is the unique integer $\floor x$ such that:

$\floor x \le x < \floor x + 1$

Hence:

$0 \le x - \floor x < 1$

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

 $\ds H_n - \ln n - \gamma$ $=$ $\ds H_n - \ln n - \lim_{k \mathop \to +\infty} \paren {H_k - \ln k}$ Definition of Euler-Mascheroni Constant and Existence of Euler-Mascheroni Constant $\ds$ $=$ $\ds 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \ln n - \lim_{k \mathop \to +\infty} \paren {1 + \int_1^k \dfrac {\floor u} {u^2} \rd u - \ln k}$ Integral Expression of Harmonic Number $\ds$ $=$ $\ds \int_1^n \dfrac {\floor u} {u^2} \rd u - \int_1^n \dfrac 1 u \rd u - \lim_{k \mathop \to +\infty} \paren {\int_1 ^k \dfrac {\floor u} {u^2} \rd u - \int_1^k \dfrac 1 u \rd u }$ Definition of Real Natural Logarithm $\ds$ $=$ $\ds \lim_{k \mathop \to +\infty} \paren {-\int_n^k \dfrac {\floor u} {u^2} \rd u + \int_n^k \dfrac 1 u \rd u }$ Sum of Integrals on Adjacent Intervals for Continuous Functions $\ds$ $=$ $\ds \lim_{k \mathop \to +\infty} \paren {\int_n^k \dfrac {u - \floor u} {u^2} \rd u }$ Integral Operator is Linear $\ds$ $<$ $\ds \lim_{k \mathop \to +\infty} \paren {\int_n^k \dfrac 1 {u^2} \rd u }$ Since $0 \le x - \floor x < 1$ $\ds$ $=$ $\ds \intlimits {-\dfrac 1 u} n \infty$ Primitive of Power $\ds$ $=$ $\ds \paren {0 - \paren {-\dfrac 1 n} }$ $\ds$ $=$ $\ds \dfrac 1 n$

From Existence of Euler-Mascheroni Constant Proof 1, we have:

$\ds \Delta_n = \sum_{k \mathop = 1}^n \dfrac 1 k - \int_1^n \dfrac 1 x \rd x$

is decreasing and bounded below by zero.

Therefore:

$H_n - \ln n \ge 0$

Therefore:

$\forall n \in \N_{>0} : 0 \le \size {H_n - \ln n - \gamma} < \dfrac 1 n$

$\blacksquare$

## Source of Name

This entry was named for Leonhard Paul Euler.