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body/woche4/index.tex %% |____ body/woche4/A1.tex +%% |____ body/woche4/A2.tex +%% |____ body/woche4/A3.tex %% |____ body/woche4/A4.tex %% |____ back/index.tex %% |____ back/sources.bib @@ -117,6 +119,7 @@ dvipsnames, }{xcolor} +\usepackage{makecmds} % need for \provideenvironment \usepackage{inputenc} \usepackage{babel} \usepackage{geometry} @@ -306,6 +309,9 @@ \providecommand{\powerSet}{} \renewcommand{\powerSet}[1]{\mathop{\powerset}(#1)} +\providecommand{\blockCases}{} +\renewenvironment{blockCases}[2]{\left\{\begin{array}[#1]{#2}}{\end{array}\right.} + \newcommand{\fieldK}[0]{\mathbb{K}} \newcommand{\reals}[0]{\mathbb{R}} \newcommand{\complex}[0]{\mathbb{C}} @@ -1069,7 +1075,7 @@ Per Definition gilt $\norm{f}_{\infty} \in \reals$ gdw. $f$ beschränkt ist. \begin{schattierteboxdunn} \begin{claim} -\setblocklabel{claim:1:ex:2.1:raj-analysis-ii-notes} +\setblocklabel{claim:1:ex:3.1:raj-analysis-ii-notes} Angenommen, so dass $(f_{n})_{n\in\naturals}$ genüge folgender Eigenschaft @@ -1238,6 +1244,463 @@ Per Definition gilt $\norm{f}_{\infty} \in \reals$ gdw. $f$ beschränkt ist. %% ********** END OF FILE: body/woche4/A1.tex ********** +\setcounterafter{section}{2} + +%% ******************************************************************************** +%% FILE: body/woche4/A2.tex +%% ******************************************************************************** + +\let\oldsectionname\sectionname +\def\sectionname{Aufgabe} +\section[]{} +\let\sectionname\oldsectionname +\label{sec:6} + +\begin{schattierteboxdunn} +\begin{claim} +\setblocklabel{claim:1:ex:3.2:raj-analysis-ii-notes} + Seien $\alpha \in (0, \infty)$. + Dann + $\displaystyle\sum_{k=1}^{\infty}k^{-\alpha} < \infty$ + gdw. $\alpha > 1$. +\end{claim} +\end{schattierteboxdunn} + + \begin{beweis} + Sei $N\in\naturals$ mit $N > 1$. + Dann + + \begin{maths}[mc]{rcl} + \eqtag[eq:1:\beweislabel] + \displaystyle\sum_{k=1}^{N}k^{-\alpha} + &= &k^{-N} + \displaystyle\sum_{k=1}^{N - 1} + \displaystyle\int_{t=k}^{k + 1} k^{-\alpha} \dee t\\ + &\geq &0 + \displaystyle\sum_{k=1}^{N - 1} + \displaystyle\int_{t=k}^{k + 1} t^{-\alpha} \dee t + = \displaystyle\int_{t=1}^{N} t^{-\alpha} \dee t\\ + \end{maths} + + und + + \begin{maths}[mc]{rcl} + \eqtag[eq:2:\beweislabel] + \displaystyle\sum_{k=1}^{N}k^{-\alpha} + &= &1 + \displaystyle\sum_{k=2}^{N} \displaystyle\int_{t=k-1}^{k} k^{-\alpha} \dee t\\ + &\leq &1 + \displaystyle\sum_{k=2}^{N} + \displaystyle\int_{t=k-1}^{k} t^{-\alpha} \dee t\\ + &= &1 + \displaystyle\int_{t=1}^{N} t^{-\alpha} \dee t.\\ + \end{maths} + + Laut Skript (siehe \cite[\S{}11.3,~Bsp.~(c)]{pogorzelskiVLSkript}) + wissen wir nun, dass + + \begin{maths}[mc]{rcl} + \eqtag[eq:3:\beweislabel] + \displaystyle\int_{t=1}^{\infty} t^{-\alpha} \dee t + &= &\begin{blockCases}{mc}{rcl} + +\infty &: &\alpha \leq 1\\ + \frac{1}{1-\alpha} &: &\alpha > 1\\ + \end{blockCases}.\\ + \end{maths} + + Folglich gelten + + \begin{maths}[mc]{rcl} + \displaystyle\sum_{k=1}^{\infty}k^{-\alpha} + &= &\displaystyle\limsup_{N \to \infty}\displaystyle\sum_{k=1}^{N}k^{-\alpha}\\ + &\eqcrefoverset{eq:2:\beweislabel}{\leq} + &\displaystyle\limsup_{N \to \infty}\displaystyle\int_{t=1}^{N} t^{-\alpha} \dee t + 1\\ + &= &\displaystyle\displaystyle\int_{t=1}^{\infty} t^{-\alpha} \dee t + 1\\ + &\eqcrefoverset{eq:3:\beweislabel}{=} + &\frac{1}{1-\alpha} + 1 + < \infty,\\ + \end{maths} + + falls $\alpha > 1$, und + + \begin{maths}[mc]{rcl} + \displaystyle\sum_{k=1}^{\infty}k^{-\alpha} + &= &\displaystyle\limsup_{N \to \infty}\displaystyle\sum_{k=1}^{N}k^{-\alpha}\\ + &\eqcrefoverset{eq:1:\beweislabel}{\geq} + &\displaystyle\limsup_{N \to \infty}\displaystyle\int_{t=1}^{N} t^{-\alpha} \dee t\\ + &= &\displaystyle\int_{t=1}^{\infty} t^{-\alpha} \dee t\\ + &\eqcrefoverset{eq:3:\beweislabel}{=} + &+\infty,\\ + \end{maths} + + falls $0 < \alpha \leq 1$. + Darum gilt die Behauptung. + \end{beweis} + +%% ********** END OF FILE: body/woche4/A2.tex ********** + +\setcounterafter{section}{3} + +%% ******************************************************************************** +%% FILE: body/woche4/A3.tex +%% ******************************************************************************** + +\let\oldsectionname\sectionname +\def\sectionname{Aufgabe} +\section[]{} +\let\sectionname\oldsectionname +\label{sec:7} + +Für diese Aufgaben brauchen wir zunächst einmal Lemma, um unsere Arbeit zu erleichtern. + +\begin{lemm} +\setblocklabel{lemm:uneigentlich:ex:3.3:raj-analysis-ii-notes} + Sei $a\in\reals$ und sei ${h:[a,\infty)\to\reals}$ eine Funktion + mit ${h(t) \longrightarrow 0}$ für ${t \longrightarrow +\infty}$. + Sei außerdem + $(T_{n})_{n} \subseteq [a,\infty)$ + eine monoton wachsende Folge mit ${T_{n} \longrightarrow \infty}$ + und so, + dass $(T_{n+1}-T_{n})_{n}$ beschränkt ist. + Dann ist + ${\displaystyle\int_{a}^{\infty} h \dee t}$ + konvergent + gdw. + $(\displaystyle\int_{a}^{T_{n}} h \dee t)_{n}$ + konvergent ist. +\end{lemm} + + \begin{einzug}[\mytab][\mytab] + \begin{beweis} + Offensichtlich gilt die \enclosedquote{nur dann wenn}-Richtung. + Für die \enclosedquote{wenn}-Richtung, + sei angenommen, $(\displaystyle\int_{a}^{T_{n}} h \dee t)_{n}$ konvergiert + mit Grenzwert $I \in \reals$. + Setze $C \in (0,\infty)$ mit $\sup_{n}T_{n+1}-T_{n} \leq C$. + + Sei $\eps > 0$. + Sei $N$ ein genügend großer Index + mit $\abs{\int_{a}^{T_{n}} h \dee t - I} < \frac{\eps}{2}$ + für alle $n \geq N$. + Da $f$ gegen $+\infty$ verschwindend ist, + existiert ein $\tilde{T}\in[a,\infty)$, + so dass $\abs{f(\cdot)} \leq \frac{\eps}{2C}$ + auf $[\tilde{T},\infty]$. + Sei nun $T \in [\max\{T_{N},\tilde{T}\},\infty)$ beliebig. + Da $T \geq T_{N}$ und ${(T_{n})_{n}\longrightarrow +\infty}$ monoton, + existiert ein $n\geq N$ so, dass $T \in [T_{n},T_{n+1}]$. + Da $n \geq N$ und $[T_{n},T]\subseteq[\tilde{T},\infty)$ + erhält man die Abschätzung + + \begin{maths}[mc]{rcl} + \absLong{ + \displaystyle\int_{a}^{T} h(t) \dee t + - + I + } + &\leq + &\absLong{ + \displaystyle\int_{a}^{T} h(t) \dee t + - + \displaystyle\int_{a}^{T_{n}} h(t) \dee t + } + + \absLong{ + \displaystyle\int_{a}^{T_{n}} h(t) \dee t + - + I + }\\ + &< &\absLong{ + \displaystyle\int_{T_{n}}^{T} h(t) \dee t + } + + \frac{\eps}{2}\\ + &< &\displaystyle\int_{T_{n}}^{T} + \underbrace{ + \abs{h(t)} + }_{\leq \frac{\eps}{2C}} + \dee t + + \frac{\eps}{2}\\ + &\leq &(T-T_{n})\frac{\eps}{2C} + + \frac{\eps}{2}\\ + &\leq &(T_{n+1}-T_{n})\frac{\eps}{2C} + + \frac{\eps}{2}\\ + &\leq &C\frac{\eps}{2C} + + \frac{\eps}{2} + =\eps.\\ + \end{maths} + + Darum gilt für genügend großes + $T \in [a,\infty)$ + (in Abhängigkeit von $\eps$), + dass $\abs{ + \displaystyle\int_{a}^{T} h(t) \dee t + - + I + } < \eps$. + Da dies für alle $\eps > 0$ gilt + ist + $(\displaystyle\int_{a}^{T} h \dee t)_{T\in[a,\infty)}$ + konvergent + (mit Grenzwert $I$). + Also konvergiert $\displaystyle\int_{a}^{\infty} h \dee t$. + \end{beweis} + \end{einzug} + +\begin{enumerate}{\bfseries (a)} + %% A3(a) + \item + \begin{schattierteboxdunn} + \begin{claim} + \setblocklabel{claim:1:ex:3.3a:raj-analysis-ii-notes} + Sei $f:\reals\ohne\{0\}\to\reals$ + definiert durch $f(x)=\frac{\sin(x)}{x}$. + Dann konvergiert + $\displaystyle\int_{x=1}^{\infty} f \dee x$. + \end{claim} + \end{schattierteboxdunn} + + \begin{beweis} + Für $c \in (0, \infty)$ beobachte, dass + + \begin{maths}[mc]{rcl} + \eqtag[eq:1:\beweislabel] + \absLong{ + \displaystyle\int_{c}^{c + 2\pi} + \frac{\sin(x)}{x} + \dee x + } + &= &\absLong{ + \displaystyle\int_{c}^{c + 2\pi} + \frac{\sin(x)}{x} + \dee x + - + \displaystyle\int_{c}^{c + 2\pi} + \frac{\sin(x)}{c} + \dee x + }\\ + &&\text{da $\int_{c}^{c + 2\pi}\sin(x)\dee x = [-\cos(x)]_{c}^{c+2\pi}=0$}\\ + &= &\absLong{ + \displaystyle\int_{c}^{c + 2\pi} + \sin(x)\cdot(\frac{1}{x} - \frac{1}{c}) + \dee x + }\\ + &\leq &\displaystyle\int_{c}^{c + 2\pi} + \underbrace{ + \abs{\sin(x)} + }_{\leq 1} + \underbrace{ + \abs{\frac{1}{x} - \frac{1}{c}} + }_{\substack{ + \leq \frac{1}{c} - \frac{1}{c + 2\pi}\\ + = \frac{2\pi}{c(c + 2\pi)} + }} + \dee x\\ + &\leq &2\pi \cdot \frac{2\pi}{c(c + 2\pi)} + \leq (\frac{2\pi}{c})^{2}.\\ + \end{maths} + + Da $f$ offensichtlich gegen $+\infty$ verschwindend ist, + laut \Cref{lemm:uneigentlich:ex:3.3:raj-analysis-ii-notes} + reicht es aus zu zeigen, + dass + $(\int_{1}^{2\pi n}f\dee x)_{n\in\naturals}$ konvergiert. + + Sei hiefür $\eps > 0$ beliebig. + Da $\sum_{k=1}^{\infty}\frac{1}{k^{2}}<\infty$, wissen wir, + dass ein $N\in\naturals$ existiert + so dass + $\abs{ + \sum_{k=1}^{n_{1}} \frac{1}{k^{2}} + - + \sum_{k=1}^{n_{2}} \frac{1}{k^{2}} + } < \frac{\eps}{3}$ + für alle $n_{1},n_{2} \geq N$. + \OE wähle $N > \frac{3}{\eps}$. + Für $n_{1},n_{2} \geq N$ berechnen wir daher: + + \begin{maths}[mc]{rcl} + \absLong{ + \displaystyle\int_{1}^{2\pi n_{1}} f \dee x + - + \displaystyle\int_{1}^{2\pi n_{2}} f \dee x + } + &= + &\absLong{ + \displaystyle\int_{2\pi \min\{n_{1},n_{2}\}}^{2\pi \max\{n_{1},n_{2}\}} f \dee x + }\\ + &= + &\absLong{ + \displaystyle\sum_{k=\min\{n_{1},n_{2}\}}^{\max\{n_{1},n_{2}\}-1} + \displaystyle\int_{2\pi k}^{2\pi (k+1)} f \dee x + }\\ + &= + &\displaystyle\sum_{k=\min\{n_{1},n_{2}\}}^{\max\{n_{1},n_{2}\}-1} + \absLong{ + \displaystyle\int_{2\pi k}^{2\pi (k+1)} f \dee x + }\\ + &\eqcrefoverset{eq:1:\beweislabel}{\leq} + &\displaystyle\sum_{k=\min\{n_{1},n_{2}\}}^{\max\{n_{1},n_{2}\}-1} + (\frac{2\pi}{2\pi k})^{2}\\ + &= + &\absLong{ + \displaystyle\sum_{1}^{n_{1}} \frac{1}{k^{2}} + - + \displaystyle\sum_{1}^{n_{2}} \frac{1}{k^{2}} + } + < \eps.\\ + \end{maths} + + Es folgt, dass + $(\int_{1}^{2\pi n}f\dee x)_{n\in\naturals}\subseteq\reals$ + eine Cauchy-Folge ist und wegen Vollständigkeit von $\reals$ konvergent. + Laut \Cref{lemm:uneigentlich:ex:3.3:raj-analysis-ii-notes} + existiert damit $\int_{1}^{\infty} f \dee x$. + \end{beweis} + + %% A3(b) + \item + \begin{schattierteboxdunn} + \begin{claim} + \setblocklabel{claim:1:ex:3.3b:raj-analysis-ii-notes} + $\int_{x=1}^{\infty} \abs{\frac{\sin(x)}{x}} \dee x$ + divergiert gegen $+\infty$. + \end{claim} + \end{schattierteboxdunn} + + \begin{beweis} + \end{beweis} + + %% A3(c) + \item + \begin{schattierteboxdunn} + \begin{claim} + \setblocklabel{claim:1:ex:3.3c:raj-analysis-ii-notes} + Sei $g:\reals\ohne\{0\}\to\reals$ + definiert durch $g(x)=\frac{\sin(x)}{x}$. + Dann divergiert + $\displaystyle\int_{x=1}^{\infty} g \dee x$ + gegen $+\infty$. + \end{claim} + \end{schattierteboxdunn} + + Um dies zu beweisen brauchen wir ein kleines Zwischenresultat. + + \begin{lemm} + \setblocklabel{lemm:1:ex:3.3c:raj-analysis-ii-notes} + Für $k, c \in [0, \infty)$ + mit $k \geq 1$ + ist + ${I_{k,c} \coloneq \displaystyle\int_{x=1}^{\infty} \frac{\sin(kx - c)}{x} \dee x}$ + konvergent. + \end{lemm} + + \begin{einzug}[\mytab][\mytab] + \begin{beweis} + Schreibe + ${I_{k,c}(T) \coloneq \displaystyle\int_{x=1}^{T} \frac{\sin(kx - c)}{x} \dee x}$ + für $T \in [1, \infty)$. + Wegen Aufgabe 3(a) wissen wir bereits, + dass + ${I_{1,0}(T) \longrightarrow I_{1,0} \in \reals}$ + für ${T \longrightarrow +\infty}$. + Für $T \in [1, \infty)$ gilt + + \begin{maths}[mc]{rcl} + I_{k,c}(T) + &= + &\displaystyle\int_{1}^{T} + \frac{\sin(kx - c)}{kx-c + c} + \cdot u^{\prime} + \dee u\\ + &&\text{mit $u(x) \coloneq kx - c$}\\ + &= + &\displaystyle\int_{k + c}^{kT + c} + \frac{\sin(u)}{u + c} + \dee u\\ + &= + &\displaystyle\int_{1}^{kT + c} + \frac{\sin(u)}{u} + \dee u + -\underbrace{ + \displaystyle\int_{1}^{kT + c} + \sin(u)\cdot(\frac{1}{u}-\frac{1}{u+c}) + \dee u + }_{\eqcolon D_{c}(kT + c)}\\ + &&-\underbrace{ + \displaystyle\int_{1}^{k + c} + \frac{\sin(u)}{u + c} + \dee u + }_{\eqcolon E_{k,c}}\\ + &= &I_{1,0}(kT + c) + D_{c}(kT + c) + E_{k,c}.\\ + \end{maths} + + Wegen Stetigkeit von ${[1,k+c] \ni x \mapsto \frac{\sin(u)}{u + c}}$, + ist diese Funktion Riemann-integrierbar. + Also ist $E_{k,c} \in \reals$ wohldefiniert. + Wie oben wissen wir, dass + ${T \longrightarrow +\infty}$ + $\Rightarrow$ + ${kT + c \longrightarrow +\infty}$ + $\Rightarrow$ + ${I_{1,0}(kT+c) \longrightarrow I_{1,0} \in \reals}$. + Darum, um die Konvergenz von $I_{k,c}$ zu zeigen, + müssen wir lediglich die Konvergenz von + $(D_{c}(kT + c))_{T\in[1,\infty)}$ + zeigen. + + Nebenrechnung: + $\abs{\sin(u)\cdot(\frac{1}{u}-\frac{1}{u+c})} + =\abs{\sin(u)}\cdot\frac{c}{u(u+c)} + \leq \frac{c}{u^{2}}$ + für $u\in[1,\infty)$. + Da $\int_{u=1}^{\infty} \frac{c}{u^{2}} \dee u$ existiert, + ist somit der Bertrag der stetigen Funktion + ${[1,\infty)\ni u \mapsto \sin(u)\cdot(\frac{1}{u}-\frac{1}{u+c})}$ + uneigentlich Riemann-integrierbar. + Folglich ist diese Funktion (ohne Betrag) + uneigentlich Riemann-integrierbar. + Insbesondere ist $(D_{c}(T))_{T\in[1,\infty)}$ + und somit auch $(D_{c}(kT + c))_{T\in[1,\infty)}$ + konvergent. + + Darum konvergiert $(I_{k,c}(T))_{T\in[1,\infty)}$. + \Dh $\displaystyle\int_{x=1}^{\infty} \frac{\sin(kx - c)}{x} \dee x$ existiert. + \end{beweis} + \end{einzug} + + \def\beweislabel{claim:1:ex:3.3c:raj-analysis-ii-notes} + Jetzt können wir mit dem Beweis von \Cref{\beweislabel} fortsetzen. + + \begin{beweis}[von \Cref{\beweislabel}] + Sei $T \in [1,\infty)$. + Schreibe ${J(T) \coloneq \int_{x=1}^{T} g \dee x}$. + Für $x\in[1,\infty)$ gilt + $% + g(x) = \frac{\sin^{2}(x)}{x} + = \frac{1}{2}(\frac{1}{x} - \frac{\cos(2x)}{x}) + = \frac{1}{2}(\frac{1}{x} - \frac{\sin(\frac{\pi}{2} - 2x)}{x}) + = \frac{1}{2}(\frac{1}{x} + \frac{\sin(2x - \frac{\pi}{2})}{x}) + = \frac{1}{2}(\frac{1}{x} + h_{2,\frac{\pi}{2}}(x)) + $. + Darum gilt + + \begin{maths}[mc]{rcl} + J(T) + &= + &\frac{1}{2} + \displaystyle\int_{x=1}^{T}\frac{1}{x}\dee x + + + \frac{1}{2} + \displaystyle\int_{x=1}^{T}h_{2,\frac{\pi}{2}}(x) \dee x\\ + &= &\frac{1}{2}\log(T) - \frac{1}{2}I_{2,\frac{\pi}{2}}(T).\\ + \end{maths} + + Da + ${(I_{2,\frac{\pi}{2}}(T))_{T\in[1,\infty)} \longrightarrow I_{2,\frac{\pi}{2}} \in \reals}$ + (siehe \Cref{lemm:1:ex:3.3c:raj-analysis-ii-notes}) + und + ${(\log(T))_{T\in[1,\infty)} \longrightarrow +\infty}$, + folgt ${(J(T))_{T\in[1,\infty)} \longrightarrow +\infty}$. + \Dh $\displaystyle\int_{x=1}^{\infty} g \dee x = +\infty$. + \end{beweis} +\end{enumerate} + +%% ********** END OF FILE: body/woche4/A3.tex ********** + \setcounterafter{section}{4} %% ******************************************************************************** @@ -1248,7 +1711,7 @@ Per Definition gilt $\norm{f}_{\infty} \in \reals$ gdw. $f$ beschränkt ist. \def\sectionname{Aufgabe} \section[]{} \let\sectionname\oldsectionname -\label{sec:6} +\label{sec:8} \begin{enumerate}{\bfseries (a)} \setcounterafter{enumi}{2} @@ -1484,7 +1947,7 @@ Otto Forster. \newblock {Grundkurs Mathematik}. Springer Spektrum, Wiesbaden, 12 edition, 2016. -\bibitem[Pog 2]{pogorzelski} +\bibitem[Pog 2]{pogorzelskiVLSkript} Felix Pogorzelski. \newblock {Vorlesungsskript: Analysis I--II}, 2021--2. \newblock {basierend auf dem Skript von Daniel Lenz 2013--14 + 2020--21}.