Euler's Integral Theorem/Proof 2
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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
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:
- $\forall n \in \N_{>0} : 0 \le \size {H_n - \ln n - \gamma} < \dfrac 1 n$
$\blacksquare$