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zYsY`05f^AQ-t+j83SK8YNglS9^_K`W?ucWq==0itS5hz1_&9t@dX%31+>>J9ae9}& zhm@hZ{D{(W-!c1dQmcddUAQ0He!=)Wa7dLwa=0>20^7{n(dq97zA2a=JI z6OoqVmzR|l5Crkc2nj*x|E~$-|8Mi`NX5#@+RcU@3`U$8qH@Anl7^jQrOEqtGez9( z8+`J=9(($TwEfLUwOc;4(M;BhD<#2VWFW+}WmJj?4LN1)$y=@_tck~X3jCTmX4O(2 z0VKq{@XhQ6j)GP_chaqQD0VaM4l^&*mQIFo-31si*^z0f-~UXOPzsdpJ4ylY;!#`w zgnq#z4Im=$p>2TUa|4Pfs1!+p7l8*)+@Giw zXGSGODG>EWq8ele7|B7a4+4+jib}n(lxEulh;^!s;dlhnj$wHOOPB%Y(653~*68<8 z9yZ=Z@@(P{)M&Blrmyiv;c+C{sVzmfwmGB=(>r<0aL*^>m8d!^iZMGmjb$wi8u08z zFU%D;WskrM`k{24#uAdq+v4f)iQ<&7 z(qi4vAe#!MktV-lHOsBk5}$;kCnCX!)@ z67@pEUijb_JEAB1A)S_xiK;?TX3KC@D`q!Sfs~XnA=^^z?yq8M-6?nk`n!ykMNCeX zj80c_D(|A39ym9S8zb(v3Gl2LY5iin?%6ix4\frac{12}{12\cdot12n^2+12\cdot12n+24n+13} \\ + &= \frac{12}{12\cdot12n^2+12\cdot14n+13} + = \frac{12}{(12n+1)(12n+13)}\\ + &= \frac{1}{12n+1}-\frac{1}{12(n+1)+1} + \end{align*} + Setzen wir beide Ungleichungen zusammen, so erhalten wir + \[ + \frac{1}{12n+1}-\frac{1}{12(n+1)+1} < d_n-d_{n+1} < \frac{1}{12n}-\frac{1}{12(n+1)} + \] + Aus + \begin{align*} + &\left(d_n-\frac{1}{12n}\right)-\left(d_{n+1}-\frac{1}{12(n+1)}\right)\\ + <\quad0\quad<\quad&\left(d_n-\frac{1}{12n+1}\right)-\left(d_{n+1}-\frac{1}{12(n+1)+1}\right) + \end{align*} + folgt, dass die Folge $\left\{d_n-\frac{1}{12n+1}\right\}$ monton fallend, die Folge $\left\{d_n-\frac{1}{12n}\right\}$ monoton wachsend ist. Beide Folgen haben dabei den Grenzwert $c:=\lim d_n$. + \begin{align*} + d_n-\frac{1}{12n}\quad&<\quad c\quad<\quad d_n-\frac{1}{12n+1}\\ + c+\frac{1}{12n+1}\quad&<\quad d_n\quad<\quad c+\frac{1}{12n} + \end{align*} + Aus unserem Start + \[ + d_n = \ln{n!}-\left(n+\frac{1}{2}\right)\ln{n+n} + \] + folgt nun + \[ + e^{d_n} = {n!}\left(\frac{n}{e}\right)^{-1}n^{-\frac{1}{2}} + \] + An dieser Stelle notieren wir + \begin{align*} + e^{c+\frac{1}{12n+1}} &< e^{d_n} < e^{c+\frac{1}{12n}} \\ + e^c e^{\frac{1}{12n+1}} &< n! \left(\frac{n}{e}\right)^{-n} n^{-\frac{1}{2}} < e^c e^{\frac{1}{12n}} \\ + e^c \sqrt{n} \left(\frac{n}{e}\right)^n e^{\frac{1}{12n+1}} &< n! < e^c \sqrt{n} \left(\frac{n}{e}\right)^n e^{\frac{1}{12n}} + \end{align*} + Es bleibt zu zeigen, dass + \[ + e^c=\sqrt{2\pi} + \] + Wir führen eine Hilfsrechnung über + \[ + I_n:=\int_{0}^{\pi/2}{sin^n x dx} + \] + Es gilt + \begin{align*} + I_n &= \int_{0}^{\pi/2} \sin^{n-1} x \cdot \sin x \, dx \\ + &= \left. \sin^{n-1} x \cdot (-\cos x) \right|_{0}^{\pi/2} - \\ + &\quad \int_{0}^{\pi/2} (n-1) \sin^{n-2} \cos x \cdot (-\cos x) \, dx \\ + &= (n-1) \int_{0}^{\pi/2} \sin^{n-2} x (1 - \sin^2 x) \, dx \\ + &= (n-1)(I_{n-2} - I_n) \\ + I_n &= \frac{n-1}{n} I_{n-2} \\ + I_0 &= \frac{\pi}{2} \\ + I_1 &= 1 \\ + I_2 &= \frac{1}{2} I_0 = \frac{1}{2} \cdot \frac{\pi}{2}\\ + I_{2n} &= \frac{1\cdot3\cdot5\dots(2n-1)}{2\cdot4\cdot6\dots(2n)}\frac{\pi}{2}\\ + I_{2n} &= \frac{2\cdot4\cdot6\dots(2n)}{1\cdot3\cdot5\dots(2n+1)} + \end{align*} + Weiterhin folgt + \begin{align*} + && 0 &< x < \frac{\pi}{2} \\ + &\Rightarrow & 0 &< \sin x < 1 \\ + &\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 \\ + &\Rightarrow & \lim \ & \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 \\ + &\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 $\lim_{n\rightarrow\infty}{\alpha_n}=\lim_{n\rightarrow\infty}{\beta_n}=0$: + \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} + \end{align*} + Einsetzen liefert schlussendlich + \[ + e^c=\sqrt{2\pi}\qquad\Box + \] + \subsection{Dreiecksform bilden}\label{dreieck} Genutzt werden sowohl Arme als auch Füße. Bilden Sie mit den Körpern aller Spieler ein Dreieck aus dem Stand. Dazu können Sie folgende Formel nutzen: Sei dazu $\text{dgw}(\psi)$ die Funktion, @@ -83,11 +406,17 @@ $k=\begin{cases*}n/2 &\text{Für } n\%2 = 0\\n/2+1&\text{sonst.}\end{cases*}$ \begin{align*} \text{dgw}(\psi): \mathbb{R}^n &\rightarrow \mathbb{R},\\ - x &\mapsto \text{det}(A^{-1}) + 2 \cdot e_{k}\cdot+\textbf{0}+A\cdot \mathds{1}_n^t\cdot A^t + \begin{pmatrix*} + x &\mapsto \text{det}(A^{-1}) + n! + 2 \cdot e_{k}\cdot+\textbf{0}+A\cdot \mathds{1}_n^t\cdot A^t + \begin{pmatrix*} 1&2&3&...\\ 2&4&7&...\\ 3&7&14&...\\ ...&...&...&... + \end{pmatrix*}\\ + & \approx \text{det}(A^{-1}) + \sqrt{2\pi n}\left(\frac{n}{e}\right)^n + 2 \cdot e_{k}\cdot+\textbf{0}+A\cdot \mathds{1}_n^t\cdot A^t + \begin{pmatrix*} + 1&2&3&...\\ + 2&4&7&...\\ + 3&7&14&...\\ + ...&...&...&... \end{pmatrix*} \end{align*} Bitte beachten Sie, dass das Ergebnis eine orthogonale obere Dreiecksmatrix sein muss. Sollte dies nicht der Fall sein, haben Sie sich verrechnet.