From 1a1b806c0a1c2422eee01d64cc423a522018694d Mon Sep 17 00:00:00 2001 From: christian Date: Fri, 6 Dec 2019 12:48:44 +0100 Subject: [PATCH] add a lot --- ws2019/ana/lectures/analysis14.pdf | Bin 0 -> 171851 bytes ws2019/ana/lectures/analysis14.tex | 341 +++++++++++++++++++++++++++++ ws2019/ipi/uebungen/ipi6.pdf | Bin 73626 -> 73626 bytes ws2019/ipi/uebungen/ipi7.pdf | Bin 0 -> 20138 bytes ws2019/ipi/uebungen/ipi7.tex | 40 ++++ ws2019/la/uebungen/la8.pdf | Bin 0 -> 57035 bytes ws2019/la/uebungen/la8.tex | 34 +++ 7 files changed, 415 insertions(+) create mode 100644 ws2019/ana/lectures/analysis14.pdf create mode 100644 ws2019/ana/lectures/analysis14.tex create mode 100644 ws2019/ipi/uebungen/ipi7.pdf create mode 100644 ws2019/ipi/uebungen/ipi7.tex create mode 100644 ws2019/la/uebungen/la8.pdf create mode 100644 ws2019/la/uebungen/la8.tex diff --git a/ws2019/ana/lectures/analysis14.pdf b/ws2019/ana/lectures/analysis14.pdf new file mode 100644 index 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a_n$. + \item Ist $\sum_{n=1}^{\infty} b_n$ divergent und $a_n \in \R$ mit + $a_n \ge |b_n|$ für fast alle $n \in \N$, so ist + die reelle Reihe $\sum_{n=1}^{\infty} a_n$ divergent. Die + Reihe $\sum_{n=1}^{\infty} b_n $ heißt Minorante der Reihe + $\sum_{n=1}^{\infty} a_n$. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(i)] + \item $(s_n)_{n\in\N}, (t_n)_{n\in\N}$ Partialsummen von + $(a_n)_{n\in\N}, (b_n)_{n\in\N}$. Dann + \[ + |s_n - s_m| \le \sum_{k=m+1}^{n} |a_k| \le \sum_{k=m+1}^{n} b_k + = |t_n - t_m| + .\] $\implies (s_n)_{n \in \N}$ ist C.F.\\ + $\implies (s_n)_{n \in \N}$ konvergiert. + \item Ann. $\sum_{n=1}^{\infty} a_n $ konvergiert + $\implies \sum_{n=1}^{\infty} a_n $ konvergente Majorante + zu $\sum_{n=1}^{\infty} b_n \implies \sum_{n=1}^{\infty} b_n$ + konvergiert. Widerspruch. + \end{enumerate} +\end{proof} + +\begin{bsp} + \begin{enumerate} + \item $\sum_{n=1}^{\infty} \frac{n}{4^{n}}$. + \[ + 2^{n} = (1+1)^{n} \ge 1+n \quad \forall n + \implies \frac{n}{2^{n}} \le \frac{n}{1+n} < 1 \quad \forall n + .\] $\implies$ + \[ + \frac{n}{4^{n}} \le \frac{n}{2^{n}} \cdot \frac{1}{2^{n}} < \frac{1}{2^{n}} \quad \forall n + .\] $\implies \sum_{n=1}^{\infty} (\frac{1}{2})^{n}$ konvergente + Majorante für $\sum_{n=1}^{\infty} \frac{n}{4^{n}}$. + \item $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} } $ ist divergent, weil + $\frac{1}{\sqrt{n} } \ge \frac{1}{n}$ ($\sqrt{n} \ge 1$ + $\implies \sum_{n=1}^{\infty} \frac{1}{n}$ divergente + Minorante. +\end{enumerate} +\end{bsp} + +\begin{satz}[Quotientenkriterium] + Sei $(a_n)_{n\in\N}$ eine Folge mit $|a_n| \neq 0$ für fast alle + $n \in \N (a_n \in \mathbb{C} \text{ oder } \R)$. + \begin{enumerate}[(i)] + \item Falls ein $0 < q < 1$ mit + \[ + \frac{|a_{n+1}|}{|a_n|} \le q \quad \forall n \in \N, n \ge N_0 + ,\] dann ist $\sum_{n=1}^{\infty} a_n$ absolut konvergent. + \item Falls $\frac{|a_{n+1}|}{|a_n|} \ge 1 \quad \forall n \ge N_0$, + so ist $\sum_{n=1}^{\infty} a_n$ divergent. + \end{enumerate} +\end{satz} + +Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht sicher. + +\begin{proof} + \begin{enumerate}[(i)] + \item $\forall n \ge N_0$ gilt: + \[ + |a_n| \le q |a_{n-1}| \le \ldots \le q^{n-N_0} |a_{N_0}| + .\] $\implies \frac{|a_{N_0}}{q^{N_0}} \sum_{n=1}^{\infty} q^{n}$ + ist konvergente Majorante. + \item $|a_n| \ge |a_{n-1}| \ge \ldots \ge |a_n| \implies + (a_n)_{n\in\N}$ keine Nullfolge $\implies \sum_{n=1}^{\infty} a_n$ + divergiert. + \end{enumerate} +\end{proof} + +\begin{bsp}[Exponentialreihe] + $\forall z \in C$ : + \[ + \sum_{n=0}^{\infty} \frac{z^{n}}{n!} =: \text{exp}(z) \text{ oder } e^{z} + .\] Zahl $e := \text{exp}(1)$ + $\sum_{n=0}^{\infty} \frac{z^{n}}{n!} $ absolut konvergent, für alle + $z \in \mathbb{C}$. + \[ + a_n := \frac{z^{n}}{n!} \implies \frac{|a_{n+1}|}{|a_n|} = + \frac{\frac{|z|^{n+1}}{(n+1)!}}{\frac{|z|^{n}}{n!}} = \frac{|z|}{n+1} + .\] Für $n \ge 2 |z|$ gilt: + \[ + \frac{|a_{n+1}|}{|a_n|} = \frac{|z|}{n+1} \le \frac{1}{2} =: q < 1 + .\] +\end{bsp} + +\begin{satz}[Wurzelkriterium] + Es sei $\sum_{n=1}^{\infty} a_n$ eine Reihe + ($a_n \in \R$ oder $\mathbb{C}$ ). + \begin{enumerate}[(i)] + \item Falls $\exists \quad 0 < q < 1$ und $N_0 \in \N$ mit + \[ + \sqrt[n]{|a_n|} \le q \quad \forall n \ge N_0 + ,\] so ist $\sum_{n=1}^{\infty} a_n$ absolut + konvergent. + \item Falls $\exists N_0$ mit $\sqrt[n]{|a_n|} \ge 1 $ $\forall n \ge N_0$, so ist $\sum_{n=1}^{\infty} a_n$ divergent. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate} + \item Für $n \ge N_0$ ist $|a_n| \le q^{n} \implies$ konvergente + Majorante + \item $\sqrt[n]{|a_n|} \ge 1 \implies |a_n| \ge 1 \implies (a_n)_{n\in\N}$ keine Nullfolge $\implies$ Divergenz. + \end{enumerate} +\end{proof} + +\begin{bem} + Im Quotientenkriterium und Wurzelkriterium wird gefordert: + \[ + \frac{|a_{n+1}|}{|a_n|} \text{ bzw. } \sqrt[n]{|a_n|} \le q < 1 + .\] Die Forderung $\frac{|a_{n+1}|}{|a_n|}$ bzw. $\sqrt[n]{|a_n|} < 1$ + reicht nicht: Bsp.: $\sum_{k=1}^{\infty} \frac{1}{k}$ divergent. +\end{bem} + +\begin{satz}[Verdichtungskriterium von Cauchy] + Sei $(a_n)_{n\in\N}$, $a_n \in \R_{+}$, eine reelle, positive + monoton fallende Nullfolge. Dann gilt + \[ + \sum_{k=1}^{\infty} a_k \text{ konvergent } \iff \sum_{k=1}^{\infty} 2^{k} a_{2k} \text{ konvergent} + .\] +\end{satz} + +\begin{proof} + durch Übung. +\end{proof} + +\begin{bsp} + $\sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}$ + \begin{enumerate} + \item $\alpha \le 0$: $\left( \frac{1}{n^{\alpha}} \right)_{n \in \N} $ + keine Nullfolge $\implies \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}$ + divergent. + \item $\alpha > 0$ : + \[ + \left( \frac{1}{n^{\alpha}} \right)_{n \in \N} + .\] eine monoton fallende Nullfolge mit + \[ + \sum_{k=1}^{\infty} 2^{k} \frac{1}{(2^{k})^\alpha} + = \sum_{k=1}^{\infty} (\underbrace{2^{1-\alpha}}_{=: q})^{k} = + \sum_{k=1}^{\infty} q^{k} + .\] mit $q = 2^{1-\alpha}$ + + Falls $|q| < 1 \iff $ konvergenz \\ + d.h. $\alpha > 1 \iff $ konvergenz + \end{enumerate} +\end{bsp} + +\subsection{Umordnen von Reihen} + +\begin{definition}[Umordnung] + Sei $\sum_{n=1}^{\infty} a_n$ eine Reihe und $\tau: N \to N$ eine + bijektive Abbildung. + + Dann heißt $\sum_{n=1}^{\infty} a_{\tau(n)} = + a_{\tau(1)} + a_{\tau(2)} + \ldots$ eine Umordnung der Reihe + $\sum_{n=1}^{\infty} a_n$. +\end{definition} + +\begin{bsp} + \[ + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \text{ konvergiert} + .\] Umordnung: + \begin{align*} + 1 - \frac{1}{2} &+ \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \frac{1}{7} + - \frac{1}{6} + \underbrace{\frac{1}{9} + \frac{1}{11} + \frac{1}{13} + \frac{1}{15}}_{> 4 \frac{1}{15} > \frac{1}{4}} + - \frac{1}{8} \\ + &+ \ldots + + \underbrace{\left( \frac{1}{2^{n}+1} + \frac{1}{2^{n}+3} + \ldots + \frac{1}{2^{n} + 2^{n} - 1} \right)}_{> 2^{n-1} \frac{1}{2^{n+1} - 1} > \frac{1}{4}} - \frac{1}{2n +2} + .\end{align*} + divergiert! + + $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ konvergent, aber nicht + absolut konvergent, deshalb kann eine Umordnung die + Konvergenzeigenschaften drastisch ändern !!! +\end{bsp} + +\begin{satz}[Umordnungssatz] + Sei $\sum_{n=1}^{\infty} a_n$ eine absolut konvergente Reihe. + + Dann konvergiert jede Umordnung dieser Reihe absolut gegen + denselben Grenzwert. +\end{satz} + +\begin{proof} + Forster. +\end{proof} + +\begin{bem} + Ist $\sum_{n=1}^{\infty} a_n$ eine reelle Reihe, welche konvergiert, + aber nicht absolut, so gibt es zu jedem $c \in \R$ oder $c = \pm \infty$ + eine Umordnung $\sum_{n=1}^{\infty} a_{\tau(n)}$, welche + gegen $c \in \R$ konvergiert (oder divergiert). +\end{bem} + +\subsection{Das Cauchy-Produkt von Reihen} + +\begin{satz}[Cauchy-Produkt] + Seien $\sum_{n=0}^{\infty} a_n$ und $\sum_{n=0}^{\infty} b_n$ + absolut konvergente Reihen. Für $n \in \N_0$ sei $c_n$ definiert + durch + \[ + c_n := \sum_{k=0}^{\infty} a_k b_{n-k} = a_0b_n + a_1b_{n-1} + + \ldots + a_nb_0 + .\] Dann ist die Reihe $\sum_{n=0}^{\infty} c_n$ absolut konvergent + mit $\sum_{k=0}^{\infty} c_n = \left( \sum_{n=0}^{\infty} a_n \right) + \left( \sum_{n=0}^{\infty} b_n \right)$ +\end{satz} + +\begin{proof} + Forster. +\end{proof} + +\begin{bem} + Die Voraussetzung der absoluten Konvergenz ist wichtig: + \[ + \sum_{n=1}^{\infty} a_n, \sum_{n=1}^{\infty} b_n \text{ mit } + a_n := b_n := \frac{(-1)^{n}}{\sqrt{n+1} } + .\] konvergieren, aber ihr Cauchy-Produkt: + \[ + c_n := \sum_{k=0}^{n} a_k b_{n-k}, \quad n \in \N_0 + .\] divergiert +\end{bem} + +\begin{proof} + durch Übung. +\end{proof} + +\begin{bsp} Für $x, y \in \mathbb{C}$ gilt + \[ + \text{exp}(x+y) = \text{exp}(x)\cdot \text{exp}(y) + .\] + + \begin{proof} + \[ + \text{exp}(x) = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}, + \text{exp}(y) = \sum_{n=0}^{\infty} \frac{y^{n}}{n!} + .\] Bilde Cauchy-Produkt + \[ + .\] + \begin{align*} + c_n &:= \sum_{k=0}^{n} a_k b_{n-k} \\ + &= \sum_{k=0}^{n} \frac{x^{k}}{k!} + \cdot \frac{y^{n-k}}{(n-k)!} \\ + &= \sum_{k=0}^{n} \frac{1}{n!} \cdot \frac{n!}{k!(n-k)!} + x^{k} y^{n-k} \\ + &= \frac{1}{n!} \sum_{k=0}^{n} \binom{n}{k} x^{k}y^{n-k} + = \frac{1}{n!} (x+y)^{n} + .\end{align*} + \end{proof} +\end{bsp} + +\begin{korrolar} + \begin{enumerate}[(i)] + \item $\forall z \in \mathbb{C}$ gilt $(\exp(z))^{-1} = \exp(-z)$ + \item $\forall x \in \R$ gilt $\exp(x) > 0$ + \item $\forall n \in \Z$ ist + \[ + \exp(n) = e^{n} = \begin{cases} + e \cdot e \cdot \ldots \cdot e & n > 0 \\ + e^{-1} \cdot e^{-1} \cdot \ldots \cdot e^{-1} & n < 0 \\ + 1 & n = 0 + \end{cases} + .\] + \end{enumerate} +\end{korrolar} + +\begin{proof} + \begin{enumerate}[(i)] + \item $\exp(-z)\cdot \exp(z) = \exp(-z+z) = \exp(0) = 1$ + \item Für $x \in \R, x \ge 0$ ist $\exp(z) = 1 + x + \ldots > 0$ + + Für $x < 0 $ ist $(\exp(x))^{-1} = (\exp(-x)) > 0 \implies \exp(x) > 0$. + \item Zz.: $\exp(n) = e^{n}$ $\forall n \in \N$ + + Vollständige Induktion:\\ + $n = 1$ : $\exp(1) = e$ nach Definition\\ + $n \to n+1$ : $\exp(n+1) = \exp(n) \cdot \exp(1) =e^{n} \cdot e + = e^{n+1}$ + + Für $n \in \Z$ mit $n < 0$ gilt: + + $(\exp(n))^{-1} = \exp(-n) = e^{-n} \implies \exp(n) = e^{n}$ + \end{enumerate} +\end{proof} + +\subsection{Potenzreihen} + +\begin{definition} + Eine Potenzreihe um den Entwicklungspunkt $z_0 \in \mathbb{C}$ ist + definiert durch + \[ + \sum_{n=0}^{\infty} a_n (z - z_0)^{n}, z \in \mathbb{C} + .\] $a_n \in \mathbb{C}, \forall n \in N_0$ +\end{definition} + +\begin{bsp}[Exponentialreihe] + $\sum_{n=0}^{\infty} \frac{1}{n!} z^{n} $ mit Entwicklungspunkt + $0 \in \mathbb{C}$ konvergiert für $\forall z \in \mathbb{C}$. +\end{bsp} + +\begin{definition}[Konvergenzradius] + Zur Potenzreihe $\sum_{k=1}^{\infty} a_k(z-z_0)^{k}$ definiere + den Konvergenzradius $\rho$ durch + \[ + \rho := \frac{1}{\lim_{n \to \infty} \text{sup } \sqrt[n]{|a_n|} } + .\] +\end{definition} + +\begin{satz} + \begin{enumerate}[(i)] + \item $\sum_{n=1}^{\infty} a_n (z - z_0)^{n}$ konvergiert absolut + $\forall z \in \mathbb{C}$ mit $|z-z_0| < \rho$ + \item $\sum_{n=1}^{\infty} a_n (z-z_0)^{n} $ divergiert für $\forall z \in \mathbb{C}$ mit $|z-z_0| > \rho$ + \item für $|z-z_0| = \rho$ ist keine allgemeine Aussage möglich. + \end{enumerate} +\end{satz} + +\begin{bsp} + \[ + \underbrace{\sum_{k=1}^{\infty}}_{\text{divergent für }|x|=1} x^{k} \qquad + \underbrace{\sum_{k=1}^{\infty} \frac{x^{k}}{k}}_{\text{div für } |z} + \qquad \sum_{k=1}^{\infty} \frac{x^{k}}{k^2} + .\] $\rho$ für alle Reihen. +\end{bsp} + +\end{document} diff --git a/ws2019/ipi/uebungen/ipi6.pdf b/ws2019/ipi/uebungen/ipi6.pdf index f1a8ef9bbfa6f499fe9fd65c046b79def8e1e0a2..a6207ec4fbe499dd18469bae436e6d03a758effe 100644 GIT binary patch delta 134 zcmbQWpJmp5mWC~i_xZI<4b2S9OiZ*54Al(`)HS*Eee+XX5=&AQG+eBV3=Ay|jSQel zwsQ(FuHc;S4)Ahxl|-WiYx6b5y%as)|9 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b/ws2019/ipi/uebungen/ipi7.tex new file mode 100644 index 0000000..0eaa8e4 --- /dev/null +++ b/ws2019/ipi/uebungen/ipi7.tex @@ -0,0 +1,40 @@ +\documentclass[uebung]{../../../lecture} + +\title{IPI: Übungsblatt 7} +\author{Samuel Weidemaier, Christian Merten} + +\begin{document} + +\begin{aufgabe} + \begin{enumerate} + \item Kunden in einer Bäckerei + + FIFO: Der erste Kunde wird zu erst bedient + \item Menschen in einem Aufzug + + LIFO: Der letzte Mensch verlässt den Aufzug als erstes (bei einer Tür) + \item Lebensmittel in Ihrem Kühlschrank + + FIFO: Die ältesten Lebensmittel müssen als erstes verbraucht werden. + \item Autos auf einer Autofähre + + FIFO: Das erste Auto steht ganz vorne auf der Fähre und verlässt diese als erstes + auf der anderen Seite. + \item Druckanträge auf einen Drucker + + FIFO: Der erste Druckauftrag wird als erstes ausgeführt. + \item Besuchen der Knoten eines Baums nach dem Depth-First-Prinzip + + FIFO: Die Elemente, die als erstes dem Baum hinzugefügt wurden, also ganz oben + stehen, werden bei der Depth-First-Suche auch als erstes ausgewertet. + \item Rekursive Funktionsaufrufe + + LIFO: Der letzte Funktionsaufruf wird als erstes vollständig ausgewertet. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + +\end{aufgabe} + +\end{document} diff --git a/ws2019/la/uebungen/la8.pdf b/ws2019/la/uebungen/la8.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bc191ea8431d312d4b6475c3f802a8910aacf089 GIT binary patch literal 57035 zcmbrmWmFtkvo+kfyF+mI#@*fBU4sS!1PSi$?yf-t1PL0P;1VPd+%>rO$2=qJe$Q~% zWbWJiA&Wof>{GRC*RJYNDvL`pvM_PLQ%=nE&BJq&07)E7ZQ%L&;hE(v-n@3PBH?D? zAo=|R&n#(e=VIYR!YpZL;$k6gVdh|N0WT;B@9g4aVPX&OzBsC-;EXPU+I~*oPmuKH z_51gNW*)G`br(e;QxHS`K}h)&MTor*-ku8}eSPmf_)_)75qo2!)40Fw1h!#kfX8iYiDxx*e3SmL zvTS&;s%_ovuBkKmxsspzwQxLkd;V@y>@ky6;JKn?MfQ1VDW~X8<+TS=v=FN~jRj|W znX7mF6f@DQ!tg;pNV$|;24&8=5qW^U*Va@@zmoZH)n^%ZfDa37B<<^-!X9T08GO&v>jW)=yV=o~ncy&;V2?@chYkzoO 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\end{enumerate} +\end{aufgabe} + +\end{document}