AndradeGonzalez-wp2017/03/solutions #3/exercise-03-Solution-plik.tex
2017-11-21 14:50:45 +00:00

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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 \[\left\{\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right\}\]
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\end{enumerate}
\end{document}
%%