92 lines
2.6 KiB
TeX
92 lines
2.6 KiB
TeX
\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 \left\{ \left\|\sum_{i_1,\dots, i_m} a_{i_1,\dots,i_m} x^1_{i_1}\dots x^m_{i_m}\right\|: \|(x_i^k)_{i=1}^n \|_\infty\leq1,\ 1\leq k\leq m\right\},
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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: \[
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\operatorname{Re} z\]
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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}[Cauchy--Hadamard] The radius of convergence $R$ of the power series
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\[
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\sum_{n=0}^\infty a_n(z-z_0)^n\quad |z-z_0|<R
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\]
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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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\end{theorem}
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\begin{defn}[Prime numbers]
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A number is called prime, if it is not compound.
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\end{defn}
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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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\[
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\begin{pmatrix}
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a_{11} & a_{12}\\
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a_{21} & a_{22}
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\end{pmatrix}
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\]
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and in the text
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$
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\left(\begin{smallmatrix}
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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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