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+\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{enumerate} +\usepackage{url} +\usepackage[linkcolor=blue]{hyperref} + +\title{Lösungsskizzen Weihnachtsblatt} +\subtitle{aus der VL Analysis 1 WS 14/15} +\author{Oliver Rümpelein} +\date{} + +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\abs}[1]{\left| #1 \right|} +\renewcommand{\epsilon}{\varepsilon} +\renewcommand{\phi}{\varphi} +\newcommand{\limni}{\lim\limits_{n\to\infty}} + +\begin{document} +\maketitle + +Diese Dokument enthält in den meisten Fällen nur Skizzen und Ideen, keine vollständigen (oder formal +korrekten) Beweise. Zwar sind Lösungsskizzen nach bestem (Ge-)Wissen ausgeführt, aber ohne Gewähr auf +Richtigkeit und Verständlichkeit. (Auch beim \LaTeX{}-Satz könnte man sich mehr Mühe geben). + +Verbesserungen und Nachfragen bitte an \href{mailto:oli_r@fg4f.de}{\nolinkurl{oli_r@fg4f.de}}. + +Die aktuellste Version mit allen Verbesserungen und Anmerkungen gibt es auf +\href{https://git.f3l.de/pheerai/loes_wb}{\nolinkurl{git.f3l.de}}. + +\section*{Aufgabe 1} +\begin{enumerate}[a)] +\item + \begin{enumerate}[i)] + \item $\sqrt{x}$ ist streng monoton steigend auf $\R^+$. (Verwende $\epsilon>0, x\geq 0$ beliebig und zeige $\sqrt{x} \leq \sqrt{x+\epsilon}$ durch quadrieren). + \item Sei nun $N \in \N$ beliebig. Wähle $x_0 > N^2$. Mit der Monotonie folgt dann $\sqrt{x} > N$ für alle $x > x_0$. + \end{enumerate} +\item + Der Trick ist, folgende Erweiterung anzuwenden: + \begin{align*} + \sqrt{n+1} - \sqrt{n} &= \frac{(\sqrt{n+1} - \sqrt{n}) \cdot (\sqrt{n+1} + \sqrt{n})}{(\sqrt{n+1} + + \sqrt{n})} \\ + &= \frac{1}{(\sqrt{n+1} + \sqrt{n})} \quad \xrightarrow[\textrm{mit a)}]{n \to \infty} 0 + \end{align*} +\item + \begin{description} + \item[Fall 1:] $c=0$: Einfach ($\limni a_n = \limni 0 = 0$), monoton steigend und fallend. + \item[Fall 2:] $c\geq 1$: Sei $a=\sqrt[n]{c}-1$. Dann ist $a > -1$ und es gilt mit Bernoulli-Ugl.: + \begin{gather*} + c = (1+a)^n \geq 1+a\cdot n\\ + \Rightarrow 0 \leq a \leq \frac{c-1}{n} \xrightarrow{n \to \infty} 0 + \end{gather*} + Folge ist monoton fallend. + \item[Fall 3:] $0 1$ und die Beh. folgt mit Fall 2. + Die Folge ist monoton steigend. + \end{description} +\item + \begin{gather*} + d_n \coloneqq \sqrt[n]{n} - 1 \\ + n=(1+d_n)^n \\ + \end{gather*} + Nach dem binomischen Lehrsatz und Auslassung der meisten Summanden folgt $d_n \xrightarrow{n\to\infty} 0$. +\end{enumerate} + +\section*{Aufgabe 2} +\begin{enumerate}[a)] +\item + Konvergenz folgt, da per Induktion immer $0 < a_{n+1} < a_n$ (Beschränkt und streng monoton). + + Grenzwert: Böser Trick. Sei $a=\lim_{n\to\infty} a_n$. Nach Vorlesung gilt $\lim_{n\to \infty} a_n = + \limni a_{n+1}$, also + \begin{align*} + a &= \limni a_{n+1} = \limni a_n - \limni a_n^2 \\ + &= a - \left(\lim_{n \to \infty} a_n\right)^2 = a -a^2 \\ + &\Rightarrow a^2 = 0 \\ + &\Rightarrow a=0 + \end{align*} +\item + Wenn der Grenzwert exisitert: Ähnlich wie oben. Sei $b=\limni b_n$. Dann gilt: + \begin{align*} + b= \limni \frac{a_n}{a_{n-1}} = \limni b_n &= \limni b_{n+1} \\ + &= \limni \frac{a_{n+1}}{a_n} = \limni \frac{a_n + a_{n-1}}{a_n} \\ + &= 1 + b^{-1} + \end{align*} + und + \begin{gather*} + \frac{1+\sqrt{5}}{2} - 1 = \frac{\sqrt{5} - 1}{2} = \left(\frac{1+\sqrt{5}}{2}\right)^{-1} + \end{gather*} + Zeige nun: $\phi \coloneqq \frac{1+\sqrt{5}}{2}$ ist Grenzwert. + Via Induktion zeigt man hierfür + \begin{align*} + \abs{b_n - \phi} &= \frac{1}{b_n} \frac{1}{\phi^n} , + \end{align*} + also geht der Grenzwert gegen Null. +\end{enumerate} + +\section*{Aufgabe 3} +\begin{enumerate}[a)] +\item + Durch direkten Vergleich sieht man $0 < \sum_{n=1}^{\infty} \frac{1}{n^2+3} < \sum_{n=1}^{\infty} + \frac{1}{n^2}$ und letzteres ist konvergent. Da jeder Summand der Reihe positiv ist konvergiert die Reihe + als Folge von Partialsummen. +\item + Mit Quotientenkriterium: + \begin{align*} + \frac{\frac{(n+1)!^2}{(2n+2)!}}{\frac{n!^2}{(2n)!}} &= \frac{(n+1)^2}{4n^2 + 6n +2 }\\ + &= \frac{n^2+2n+1}{4n^2+6n+2} \xrightarrow{n \to \infty} \frac{1}{4} + \end{align*} +\item + Mit Leibnizkriterium (Abschätzen der Folge durch $\frac{1}{\sqrt{n}}$). +\end{enumerate} + +\section*{Aufgabe 4} +\begin{enumerate}[a)] +\item + Nach Aufgabe 1 konvergiert $\sqrt[n]{n}$ gegen $1$, also ist $\frac{1}{\sqrt[n]{n}}$ keine Nullfolge. + Die Reihe kann also auch nicht absolut konvergieren. +\item[b),c),d)] + Konvergieren nach Leibnizkriterium, da Nullfolgen. + b) konvergiert absolut nach Majorantenkriterium (z.B. für $n>100$ ersetze $-1000$ durch $-n^2$ und erhalte + $\frac{7}{4n^2}$, c) und d) konvergieren nicht absolut. Minorante für c): $\frac{\sqrt{n}}{n} > + \frac{1}{n}$, für d) ähnlich mittels $\sqrt{n} \rightarrow \sqrt{n-1}$. +\end{enumerate} + +\section*{Aufgabe 5} +Siehe jedes bessere Analysisbuch. Grundidee: +\begin{enumerate}[a)] +\item Abschätzen durch geometrische Reihe +\item Finden einer konvergenten Teilfolge, die keine Nullfolge ist +\item Betrachte $\frac{1}{n}$ und $\frac{1}{n^2}$). +\end{enumerate} + +\section*{Aufgabe6} +\begin{enumerate}[a)] +\item + Mittels Reihendarstellung: $\exp(x)= \sum_{n=0}^{\infty}\frac{z^n}{n!} > \frac{z^{2017}}{2017!}$, und damit: + \begin{align*} + \sum_{n=1}^{\infty} \frac{n^{2015}}{2014^n} &= \sum_{n=1}^{\infty} + \frac{n^{2015}}{\exp(n\cdot\ln(2014))} \\ + &\leq \sum_{n=1}^{\infty} \frac{n^{2015}}{\frac{(n \ln(2014))^{2017}}{2017!}} \\ + &= \sum_{n=1}^{\infty} \frac{n^{2015} 2017!}{n^{2017} \ln(2014)^{2017}} \\ + &= \sum_{n=1}^{\infty} C \cdot \frac{1}{n^2} + \end{align*} +\item + \begin{description} + \item[Fall 1:] $\alpha = 0$. Nach fallenlassen der ersten Glieder hat man $\frac{1}{n^n} < \frac{1}{n^2}$, + also Konvergenz nach Majorantenkriterium. + \item[Fall 2:] $0 < \alpha < 1$. + Wurzelkriterium (siehe Aufgabe 5), da $\alpha+\frac{1}{n} > 0$: + \begin{gather*} + \sqrt[n]{\left(\alpha + \frac{1}{n}\right)^n} = \alpha + \frac{1}{n} \xrightarrow{n \to \infty} + \alpha + \end{gather*} + und da $\alpha < 1$ existert auch ein $\alpha < \theta < 1$. + \item[Fall 3:] $ \alpha \geq 1$: + \begin{gather*} + \left(\alpha + \frac{1}{n}\right)^n \geq \alpha^n \geq 1^n + \end{gather*} + Die Reihe divergiert also nach Minorantenkriterium. + \end{description} +\end{enumerate} + +\section*{Aufgabe 7} +Folgt leicht mittels streng monoton steigend und unbeschränkt in beide Richtungen. + +\section*{Aufgabe 8} +\begin{enumerate}[a)] +\item + Siehe Blatt 12/4. +\item + Angenommen, $f$ sei L.-stetig für ein $L>0$ (OBdA $L>1$). Dann gilt mit $x=0, y > L$: + \begin{gather*} + \left| f(0) - f(y) \right| = y^2 > L*y + \end{gather*} + also Widerspruch. +\item + Wähle z.B. für $\epsilon > 0$: $\delta < \frac{\epsilon}{L}$. +\item + Grundidee: $\delta$ aus c) ist nicht von $x_0$ abhängig, nur von $\epsilon$. +\item + Teil 1 siehe Blatt 12/4, Teil 2: Ja, nach Satz 7.5 ist eine stetige Fkt. auf einer kompakten Menge (also auf + einem abgeschlossenen und beschränkten Intervall) immer gleichmäßig stetig. +\end{enumerate} + +\section*{Aufgabe 9} +Anwenden des Zwischenwertsatzes mit $a_1=0$ ($e^0 + e^{-0} = 2$) und $a_2 = 2$ ($e^2 + e^{-2} > e^2 > 2^2 =4$). + +\section*{Aufgabe 10} +\begin{enumerate}[a)] +\item + Aus $f'(x) = C$ folgt $f$ linear, der Rest ist einfach +\item + $f$ ist ganz einfach identisch zu $x \mapsto x^2$. +\end{enumerate} + +\section*{Aufgabe 11} +Ohne Gewähr: 6 Teile. +Überlegung: Wir klassifizieren einen Punkt $(x,y,z)$ im Würfel, der nicht auf einer der Ebenen liegt, durch: +\begin{align*} + (e_1,e_2,e_3)& e_1,e_2,e_3 \in \{0,1\}\\ + \intertext{wobei} + e_1 &= \left\{\begin{array}{rr} 0 & \textrm{, falls } x y + \end{array}\right. \\ + e_2 &= \left\{\begin{array}{rr} 0 & \textrm{, falls } x z + \end{array}\right. \\ + e_3 &= \left\{\begin{array}{rr} 0 & \textrm{, falls } y z + \end{array}\right. \\ +\end{align*} +Dann gibt es per se erst einmal $2^3 = 8$ mögliche Mengen , in denen +ein Punkt liegen kann wenn er nicht auf einer der Ebene liegt (identifiziert mit allen dreier-Kombinationen +von 1 und 0). Die beiden Fälle $(0,1,0)$ und $(1,0,1)$ können jedoch nicht auftreten, da sonst $x