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