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03/solutions #3/exercise-03-Solution-plik.tex

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-\documentclass{article}
-\usepackage[utf8]{inputenc}
-\usepackage{polski}
-\usepackage{amsmath}
-\begin{document}
-
-
-
-\begin{enumerate}
-\item Properly typeset the following command and properly refere to it in the text
-\[
-(\sum_{i_1,\dots,i_m} a_{i_1,\dots,i_m} ^{2m}{m+1} ^{\frac{m+1}{2m}} \leq C \sup\{ |\sum_{i_1,\dots, i_m} a_{i_1,\dots,i_m} x^1_{i_1}\dots x^m_{i_m}|: \|(x_i^k)_{i=1}^n \|_\infty\leq1,\ 1\leq k\leq m\},
-\]
-
-\item Properly typset the expression: $Re z$.
-
-\item Properly typeset indexes in the following sum:
-\[
-f(x)=\sum_{n=0, k=2}^\infty a_n^k
-\]
-
-\item Properly typeset the following theorem
-
-\noindent\textbf{Theorem 1} (Cauchy--Hadamard) \emph{The radius of convergence $R$ of the power series 
-\[
-\sum_{n=0}^\infty a_n(z-z_0)^n\ \ \ \ \ |z-z_0|<R
-\]
-can by calculated via the following formula
-\[
-\frac{1}{R}=limsup_{n\to\infty} \sqrt[n]{|a_n|}.
-\]}
-
-\noindent\textbf{Definition 2.} (Prime numbers) A number is called prime, if it is not compound. 
-
-\item Typeset the follwing matrix (display-style):
-\[
-\left\{\begin{array}{cc}
-         a_{11} & a_{12}\\
-         a_{21} & a_{22}
-       \end{array}
-     \right\}
-\]
-and in the text \[\left\{\begin{array}{cc}
-                            a_{11} & a_{12}\\
-                            a_{21} & a_{22}
-                          \end{array}
-     \right\}\]
-     Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do
-     eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut
-     enim ad minim veniam, quis nostrud exercitation ullamco laboris
-     nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in
-     reprehenderit in voluptate velit esse cillum dolore eu fugiat
-     nulla pariatur. Excepteur sint occaecat cupidatat non proident,
-     sunt in culpa qui officia deserunt mollit anim id est laborum.
-
-\end{enumerate}
-\end{document}
-                                                  
-%%