From cdcdf74dba2f029e94962bee9ee3e8037b641b67 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Tue, 14 Jul 2026 13:22:46 +0200 Subject: [PATCH] fix: Nummerierung vereinheitlich --- content/Mathematischer_Hintergrund.tex | 11 ++++++----- galton.pdf | Bin 499830 -> 499829 bytes material/ab1.pdf | Bin 125548 -> 125804 bytes material/ab1.tex | 2 ++ material/ab2.pdf | Bin 82277 -> 82345 bytes material/ab2.tex | 4 +++- material/ab3.pdf | Bin 274305 -> 302585 bytes material/ab3.tex | 1 + material/ab3_Lös.pdf | Bin 342493 -> 369801 bytes material/ab3_Lös.tex | 1 + 10 files changed, 13 insertions(+), 6 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 739509e..12f5aa6 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -318,7 +318,7 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F =&\frac{2n+1}{2}\ln(\frac{n+1}{n})-1\\ =&(2n+1)\cdot\frac{1}{2}\ln\frac{1+\frac{1}{2n+1}}{1-\frac{1}{2n1}}-1 \end{align*} - Für $-1 \sin^{2n} x > \sin^{2n+1} x \\ + &\Rightarrow & \sin^{2n+1} x &< \sin^{2n} x < \sin^{2n-1} x \\ &\Rightarrow & I_{2n-1} &> I_{2n} > I_{2n+1} \\ &\Rightarrow & \frac{I_{2n-1}}{I_{2n+1}} &> \frac{I_{2n}}{I_{2n+1}} > 1 \\ &\Rightarrow & \frac{2n+1}{2n} &> \frac{1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdots (2n-1) \cdot (2n+1)}{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots (2n) \cdot (2n)} \cdot \frac{\pi}{2} > 1 \\ @@ -416,7 +417,7 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F &\Rightarrow & \frac{\pi}{2} &= \lim \left( \frac{2 \cdot 4 \cdot 6 \cdots (2n)}{1 \cdot 3 \cdot 5 \cdots (2n-1)} \right)^2 \cdot \frac{1}{2n+1} \\ && &= \lim \frac{2^{4n} (n!)^4}{((2n)!)^2 (2n+1)} \end{align*} - Es ist also für $\lim_{n\rightarrow\infty}{\alpha_n}=\lim_{n\rightarrow\infty}{\beta_n}=0$: + Es ist also für zwei gegen $0$ konvergierende Terme $\lim_{n\rightarrow\infty}{\alpha_n}=\lim_{n\rightarrow\infty}{\beta_n}=0$: \begin{align*} n!&=e^c\sqrt{n}n^ne^{-n}e^{\alpha_n}\\ (2n)!&=e^c\sqrt{2n}2^{2n}n^{2n}e^{-2n}e^{\beta_n} @@ -542,7 +543,7 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig wobei das Lagrange-Restglied etwa $R_2(x)\sim O(x^3)$, denn \begin{align*} R_2(x)&=\frac{1}{(2+1)!}\cdot(\ln(1-\xi))^{(2+1)}\cdot(x-0)^{(2+1)}\\ - &=-\frac{1}{3(1-\xi)^3}\cdot x^3\quad\text{für ein }\xi\in(0,x)&&\text{\cref{lem:dreimalableiten}}\\ + &=-\frac{1}{3(1-\xi)^3}\cdot x^3\quad\text{für ein }\xi\in(0,x)\\ \end{align*} wobei nach der Landau-Notation gilt: $a\cdot x^n+b\in O(x^n)$ \[ diff --git a/galton.pdf b/galton.pdf index dcad2d2552e7998349df7c4d83f33ec1e35317b3..444c193802e154a17f6cb52e0433bf58b08fdf88 100644 GIT binary patch delta 69980 zcmV)BK*PWGz#jF$9*|Ujx$&;A=;RRe0B$ay{n$$&N0wrH18i2x8nB&1OGC{N>gIU+ 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a/material/ab1.tex +++ b/material/ab1.tex @@ -240,6 +240,7 @@ \end{multicols} \begin{enumerate} + \item Berechne nun nach folgender Regel: \begin{center} \textit{Die Summe des Steins ist die Summe der darüberliegenden Steine. } @@ -356,6 +357,7 @@ \end{minipage} \begin{enumerate} + \setcounter{enumi}{2} \item Zeichne in das Galtonbrett alle Wege ein, die eine Murmel nehmen kann, wenn sie auf den obersten Stab fallen gelassen wird. 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a/material/ab3_Lös.tex +++ b/material/ab3_Lös.tex @@ -234,6 +234,7 @@ \ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}} \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} + \setcounter{section}{2} \section{\curtitle} Das Galtonbrett lässt sich wir folgt formal mathematisch formulieren: \begin{definitionbox}[Modell des Galtonbretts]

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