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%%
%% An UIT Edition example
%%
%% Example 10-06-2 on page 209.
%%
%% Copyright (C) 2010 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{375.57637pt}
\AtBeginDocument{\setlength\parindent{0pt}}
\StartShownPreambleCommands
\usepackage{amsmath}
\usepackage[sans]{dsfont}
\StopShownPreambleCommands
\begin{document}
\begin{align*}
\sum_{i=n}^m a_i &= \sum_{j=n}^m a_j \tag{index renaming}\\
\sum_{i=n}^m a_i &= \sum_{i=n-l}^{m-l} a_{i+l} &\forall\, l\in\mathds{Z}{}
\tag{index shift},\\
\sum_{i=n}^m a_i+\sum_{i=m+1}^k a_i &= \sum_{i=n}^k a_i,
&\text{if } n\le m<k,\\
\sum_{i=n}^m a_i+\sum_{i=n}^m b_i &= \sum_{i=n}^m (a_i+b_i),\\
\sum_{i=n}^m c\times a_i &= c\times\sum_{i=n}^m a_i & \forall\, c\in\mathds{R}{,}\\
\sum_{i=n}^m a_i \times\sum_{j=l}^k b_j &=
\sum_{i=n}^m\sum_{j=l}^k a_i b_j
\end{align*}
\end{document}
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