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z46|?0*r_)A-` zs~#}tHoJdIzV{>Lk`5+-ITlYNOiYf+5c=Z2cRR#^G$x%~%V^|1MK^4^uafk8Y23$s@qW O3kKsdG0CXP;{F%qql0b$ literal 0 HcmV?d00001 diff --git a/ws2019/ana/lectures/analysis13.tex b/ws2019/ana/lectures/analysis13.tex new file mode 100644 index 0000000..738e8a5 --- /dev/null +++ b/ws2019/ana/lectures/analysis13.tex @@ -0,0 +1,260 @@ +\documentclass{../../../lecture} + +\begin{document} + +\subsection{Konvergenz in $\mathbb{C}$} + +Eine Folge $(z_n)_{n\in\N}$ komplexer Zahlen $(z_n \in \mathbb{C} \quad \forall n \in \N)$ konvergiert +gegen $z \in \mathbb{C}$, falls $\forall \epsilon > 0$ $\exists n_\epsilon \in \N$, s.d. $|z_n - z| < \epsilon$ $\forall n \ge n_\epsilon$ + +Genauso wird die Beschränktheit und Begriff der C.F. übertragen. + +Aus Definitionen: +\[ + |z| := \sqrt{(\text{Re}(z))^{2} + (\text{Im}(z))^{2}} +.\] und der Ungleichung: +\[ + max(|x|, |y|) \le \sqrt{x^2 + y^2} \le |x| + |y| \quad \forall x,y \in \R +.\] folgt: + +\begin{enumerate} + \item $z_n \to z, n \to \infty$ in $\mathbb{C}$ $\iff \text{Re}(z_n) \to \text{Re}(z) $ und $\text{Im}(z_n) \to \text{Im}(z)$ + \item $(z_n)_{n\in\N}$ ist eine C.F. in $\mathbb{C} \iff$ \\ + $(\text{Re}(z_n))_n$ und $\left( \text{Im}(z_n) \right)_n $ sind + C.F. in $\R$ + \item $\mathbb{C}$ ist vollständig, d.h. jede C.F. in $\mathbb{C}$ + ist konvergent. + \item Jede beschränkte Folge in $\mathbb{C}$ besitzt eine + konvergente Teilfolge. +\end{enumerate} + +\begin{bsp} + \begin{enumerate} + \item + $\frac{1+i(n+1)}{n} = \frac{1}{n} + i\left(1 + \frac{1}{n}\right) \to i$, $n \to \infty$ +\item $z_n = \left( \frac{i}{2} \right) ^{n} = \left( \frac{i}{2}, -\frac{1}{4}, -\frac{i}{8}, \frac{1}{16}, \ldots \right) $ \\ + $|z_n| = \left( \frac{1}{2} \right)^{n} \to 0$, $n \to \infty$ +\item $q^{n} \to 0$, $n \to \infty$ $\forall q \in \mathbb{C}$ mit $|q| < 1$. + \end{enumerate} +\end{bsp} + +\subsection{Unendliche Summe (,,Reihen'')} + +\begin{definition} + Sei $(a_n)_{n\in\N}$ eine Folge reeller oder komplexer Zahlen. Wir + betrachten die Folge der $n$-ten Partialsumme $(s_n)_{n \in \N}$ + definiert durch: + \[ + s_n := \sum_{k=1}^{\infty} a_k + .\] Die Reihe $\sum_{k=1}^{\infty} a_k $ konvergiert (divergiert), + wenn die Folge der Partialsummen $(s_n)_{n\in\N}$ konvergiert (divergiert). + + Im Fall von Konvergenz bezeichnet: + \[ + s_\infty := \lim_{n \to \infty} s_n = \sum_{k=1}^{n} a_k + .\] die Summe oder den Wert der Reihe. +\end{definition} + +\begin{bem} + Man kann auch Reihen $\sum_{k=l}^{\infty} a_k$ mit + $l \in \Z$ betrachten. +\end{bem} + +\begin{bsp}[Geometrische Reihe] + \[ + \sum_{k=0}^{\infty} q^{k} = 1 + q + q^2 + q^{3} + \ldots + .\] $\sum_{k=0}^{\infty} q^{k}$ konvergiert genau + für alle $ q \in \mathbb{C}$ mit $|q| < 1$ und + es gilt $\sum_{0=1}^{\infty} q^{k} = \frac{1}{1-q}$. + \label{geometrischereihe} +\end{bsp} + +\begin{proof} + Folge der Partialsummen + \[ + s_n = \sum_{k=0}^{\infty} q^{k} = \begin{cases} + \frac{1 - q^{n+1}}{1 - q} \quad q \neq 1 & q \neq 1 \\ + n + 1 & q = 1 + \end{cases} + .\] + Für $|q| < 1$ gilt $|q|^{n+1} \to 0$, $n \to \infty$ + \[ + \implies s_n = \frac{1-q^{n+1}}{1 - q} \to \frac{1}{1-q} \quad \text{für } |q| < 1 + .\] + + Bleibt zu zeigen: $\sum_{k=0}^{\infty} q^{k} $ divergent für $|q| \ge 1$. + + Angenommen $\exists q \in \mathbb{C}$ mit $|q| \ge 1$ und + $(s_n)_{n \in \N}$ konvergiert. + + Dann + \[ + |q|^{n+1} = |\underbrace{s_{n+1}}_{\to s_*} - \underbrace{s_n}_{\to s_*}| + .\] + Widerspruch, da $|q|^{n+1} \ge 1$ für $|q| \ge 1$ +\end{proof} + +\begin{lemma} + Ist $\sum_{k}^{\infty} a_k$ konvergent, dann ist $(a_k)_{k \in \N}$ eine + Nullfolge. +\end{lemma} + +\begin{proof} + $a_{n+1} = s_{n+1} - s_n \to s_{\infty} - s_{\infty} = 0$, $n \to \infty$ +\end{proof} + +\begin{bem} + Die Eigenschaft von $(a_k)_{k\in\N}$ eine Nullfolge zu sein, + reicht nicht für die Konvergenz von $\sum_{k=1}^{\infty} a_k $ aus! +\end{bem} + +\begin{bsp}[Harmonische Reihe] + \[ + \sum_{k=1}^{\infty} \frac{1}{k} \qquad \text{ist strikt divergent} + .\] + \begin{proof} + Folge der Partialsummen ist unbeschränkt: + \begin{align*} + S_{2^{m}} &= \sum_{k=1}^{2^{m}} \frac{1}{k}\\ + &= + \underbrace{1}_{\ge \frac{1}{2}} + \underbrace{\frac{1}{2}}_{\ge \frac{1}{2}} + \underbrace{\frac{1}{3} + \frac{1}{4}}_{\ge 2 \cdot \frac{1}{4} = \frac{1}{2}} + \underbrace{\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}}_{\ge 4\cdot \frac{1}{8} = \frac{1}{2}} + \ldots + + \underbrace{\frac{1}{2^{n-1}+1} + \ldots + \frac{1}{2^{m}}}_{\ge 2^{m-1} \cdot \frac{1}{2^{m}} = \frac{1}{2}} \ge \frac{m}{2} + .\end{align*} + \end{proof} +\end{bsp} + +\begin{bsp} + \[ + \sum_{k=0}^{\infty} (-1)^{k} + .\] + \begin{enumerate}[a)] + \item \[ + \sum_{k=0}^{\infty} (-1)^{k} = (1-1) + (1-1) + \ldots = + 0 + 0 + \ldots = 0 + .\] + \item + \[ + \sum_{k=0}^{\infty} (-1)^{k} = 1 + (-1+1) + (-1+1) + \ldots + = 1 + 0 + 0 = 1 + \] + \item + \[ + \sum_{k=1}^{\infty} (-1)^{k} = \frac{1}{1 -(-1)} = \frac{1}{2} + .\] + \end{enumerate} + Alles Falsch (Kostina: ,,Alles Schrot!''), weil $\sum_{k=0}^{\infty} (-1)^{k} $ divergent. +\end{bsp} + +\subsubsection{Konvergenzkriterien} +Die Konvergenz einer Reihe ist nichts anderes als die Konvergenz der +Folge der Partialsummen. + +\begin{satz} + \begin{enumerate} + \item + \[ + \sum_{k=1}^{\infty} a_k = a_{\infty} \quad \text{und} \quad \sum_{k=1}^{\infty} b_k = b_{\infty} + .\] $\implies$ + \[ + \sum_{k=1}^{\infty} (\lambda a_k + \mu b_k) = \lambda a_{\infty} + \mu b_{\infty} \quad \forall \lambda, \mu \in \mathbb{C} + .\] +\item Ist $a_k \in \R$ mit $a_k \ge 0$ $\forall k \in \N$ + dann gilt: $\sum_{k=1}^{\infty} a_k$ konvergent $\iff$ $\left( \sum_{k=1}^{n} a_k \right)_{n \in \N}$ nach oben beschränkt. + \end{enumerate} +\end{satz} + +\begin{proof} + (1) folgt aus der linearen Kombination von Folgen. \\ + (2) Falls $a_k \ge 0 \implies$ Folge $\left( \sum_{k=1}^{n} a_k \right)_{n \in \N}$ ist monoton wachsend. + + Für solche Folgen gilt: Konvergenz $\iff$ Beschränktheit +\end{proof} + +\begin{satz}[Leibniz-Kriterium] + Ist $(a_k)_{k \in \N}$ eine monoton fallende reelle Nullfolge, so + ist die alternierende Reihe: + \[ + \sum_{k=1}^{\infty} (-1)^{k} a_k = -a_1 + a_2 - a_3 + a_4 - \ldots + .\] konvergent mit folgender Abschätzung: + \[ + \left| \sum_{k=n+1}^{\infty} (-1)^{k} a_k \right| \le |a_n| \quad \forall n \in \N + .\] +\end{satz} + +\begin{proof} + Aus Voraussetzungen: $a_k \ge 0$, $s_n := \sum_{k=1}^{n} a_k$. + \[ + s_{2n+1} = s_{2n-1} + a_{2n} - a_{2n+1} \stackrel{a_{2n} \ge a_{2n+1}}{\ge } s_{2n-1} + .\] Folge mit ungeraden Indizes: $s_1 \le s_3 \le s_5 \le \ldots$. + \[ + s_{2n} = s_{2n-2} - a_{2n-1} + a_{2n} \le s_{2n-2} + .\] Folge mit geraden Indizes: $ \ldots \le s_4 \le s_2 \le s_0$ + + $s_{2n} > s_{2n+1}$ ($s_{2n+1} = s_{2n} - a_{2n+1})$ \\ + $s_{2n} - s_{2n+1} = a_{2n+1} \to 0$, $n \to \infty$ + + $s_1 \ge s_3 \le s_5 \le \ldots \le s_4 \le s_2 \le s_0$ + + $\implies [s_{2n+1}, s_{2n}]$ bilden eine Intervallschachtelung, d.h. + \[ + \exists s_{\infty} = \bigcap_{k = 1}^{\infty} [s_{2k +1}, s_{2k}] + .\] $s_{\infty} = \lim_{n \to \infty} s_{2n+1} = \lim_{n \to \infty} s_{2n}$ + \[ + s_{2n+1} \le s_{\infty} \le s_{2n} + .\] Damit gilt $\forall k \in \N$ : + \[ + 0 \le s_{\infty} - s_{2n+1} \le s_{2n} - s_{2n+1} = a_{2n+1} + .\] und + \[ + 0 \ge s_{\infty} - s_{2n} \ge s_{2n+1} - s_{2n} = - a_{2n+1} \ge - a_{2n} + .\] $\implies$ + \[ + 0 \le |s_{\infty} - s_n| \le a_n \to 0 \quad n \to \infty + .\] und + \[ + \left|\underbrace{\sum_{k=1}^{\infty} a_k}_{s_{\infty}} - \sum_{k=1}^{n} a_k \right| + = \left| \sum_{k=n+1}^{\infty} a_k\right| \le a_n + .\] +\end{proof} + +\begin{bsp}[,,Alternierende harmonische Reihe''] + \[ + \sum_{k=1}^{\infty} (-1)^{n-1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \ldots + .\] ist konvergent +\end{bsp} + +\begin{definition} + Eine Reihe $\sum_{k=1}^{\infty} a_k$ heißt absolut konvergent falls + $\sum_{k=1}^{\infty} |a_k| $ konvergiert. +\end{definition} + +\begin{bsp} + Die Reihe $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n}$ ist + konvergent nach Leibniz Kriterium, aber nicht absolut konvergent, weil + $\sum_{n=1}^{\infty} \frac{1}{n}$ divergiert. +\end{bsp} + +\begin{satz} + Aus absoluter Konvergenz einer Reihe folgt deren Konvergenz. +\end{satz} + +\begin{proof} + Sei $\sum_{k}^{} a_k $ absolut konvergent, d.h. + $\sum_{k}^{} |a_k|$ konvergiert. + \[ + s_n := \sum_{k=1}^{n} a_k, t_n := \sum_{k=1}^{n} |a_k| + .\] Dann gilt für $m, n \in \N$ mit $m \ge n$. + \[ + |s_m - s_n| = \left| \sum_{k=n+1}^{m} a_k \right| + \le \sum_{k=n+1}^{m} |a_k| = t_m - t_n = |t_m - t_n| + .\] + Da $(t_n)_{n \in \N}$ konvergiert $\implies (t_n)_{n \in \N}$ ist C.F. + + Aus $|s_m - s_n| \le |t_m - t_n| < \epsilon$ + \begin{align*} + &\implies (s_n)_{n \in \N} \text{ ist auch C.F. in $\R$ bzw. $\mathbb{C}$} \\ + &\implies \sum_{k} a_k \quad \text{konvergiert} + .\end{align*} +\end{proof} + +\end{document}