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\begin{document}

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\begin{center}
\textbf{\large Mathematik für Wirtschaftsinformatiker}\\
\end{center}
\textbf{Wintersemester 2015/2016\hfill Prof. Dr. M. Keller}\\

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\textbf{Blatt 8}\hfill % Nr des Blatts
\textbf{Abgabe 14.01.2016}\\

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\begin{itemize}
\item[(1)] Zeigen Sie: Für jedes $n\in\IN$ gilt die Ungleichung \[\sum_{k=1}^n\frac{1}{k^2}\leq 2-\frac{1}{n}.\]

\item[(2)] Zeigen Sie: Für jedes $n\in\IN$ gilt die Gleichung \[\sum_{k=n}^{2n}k=3\sum_{k=1}^n k.\] 

\item[(3)]\begin{itemize}
\item[(a)] Finden Sie alle Lösungen der Gleichung $z^3=1$ in $\IC$. Zeichnen Sie diese in die Gaußsche Zahlenebene. Tipp: Schreiben Sie $z^3=(a+ib)^3$. Was sind $\Re z^3$ und $\Im z^3$?
\item[(b)] Berechnen Sie $\Re\frac{1+i}{1-i}$ und $\Im\frac{1+i}{1-i}$. Zeichnen Sie $\frac{1+i}{1-i}$ in die Gaußsche Zahlenebene. Tipp: Erweitern Sie den Bruch geschickt.
\end{itemize}
\item[(4)]
Berechnen Sie $\Re\frac{42}{i}$ und $\Im\frac{42}{i}$. Zeichnen Sie $\frac{42}{i}$ in die Gaußsche Zahlenebene.
\end{itemize}
\end{document}