Existence of Euler-Mascheroni Constant/Proof 2
Jump to navigation
Jump to search
Theorem
The real sequence:
- $\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$
This limit is known as the Euler-Mascheroni constant.
Proof
For $n \in \N_{>0}$ let:
- $\ds \gamma_n := \sum_{k \mathop = 1}^n \frac 1 k - \ln n$
Then:
\(\ds \gamma_n\) | \(=\) | \(\ds 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \ln n\) | 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 | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds 1 - \int_1^n \dfrac {u - \floor u} {u^2} \rd u\) | Linear Combination of Definite Integrals | ||||||||||
\(\ds \) | \(\ge\) | \(\ds 1 - \int_1^n \dfrac 1 {u^2} \rd u\) | Relative Sizes of Definite Integrals as $0 \le u - \floor u < 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 n\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds 0\) |
On the other hand:
\(\ds \gamma_n - \gamma_{n + 1}\) | \(=\) | \(\ds \paren {1 - \int_1^n \dfrac {u - \floor u} {u^2} \rd u} - \paren {1 - \int_1^{n + 1} \dfrac {u - \floor u} {u^2} \rd u}\) | by $\paren 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_n^{n + 1} \dfrac {u - \floor u} {u^2} \rd u\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
\(\ds \) | \(\ge\) | \(\ds 0\) | as $u - \floor u \ge 0$ |
Thus by monotone convergence theorem, the sequence $\sequence {\gamma_n}$ converges to a limit in $\R_{\ge 0}$.
$\blacksquare$