\documentclass{article}
\newcommand{\R}{\mathbb{R}}
\usepackage[bookmarks]{hyperref}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath, amssymb}

\begin{document}

\section{Définitions}
\subsection{Somme d'une série}

\[
	\sum_{k=0}^{+\infty} u_n := \lim_{n \to \infty} \sum_{k=0}^{n} u_k
\] 

\subsection{Convergeance}
\[
	\sum_{k=0}^{+\infty} < +\infty
\] 

\subsection{Divergeance}
\[
	\sum_{k=0}^{+\infty} \not\in \R
\] 


\subsection{$n$-ième reste d'une série convergeante}
\[
	R_n := \sum_{k=n+1}^{+\infty} u_k
\] 

Et on a \[
	\sum_{k>=n+1}^{\infty} u_k \in \R
\] 

\subsection{Divergeance grossière}
\[
	\sum_{k=0}^{+\infty} u_k \text{DVG} \iff u_k \text{diverge}
\] 

\section{Théorèmes}
\subsection{Divergeance grossière}
\[
	\sum_{n\in \N}^{+\infty} u_n \in \R \implies u_n \to 0
\] 

\[
	u_n \not\to 0 \implies \sum_{n\in \N}^{+\infty} u_n \text{diverge}
\] 

\subsection{}
\[
	\text{DVG} \implies \text{diverge}
\] 

\subsection{Linéarité}
\[
	\sum_{n\in \N} u_n, \sum_{n\in \N} v_n \text{convergent} \implies \forall \lambda, \mu\in \K, \sum_{k=0}^{+\infty} \lambda u_k + \mu v_k = \lambda \sum_{k=0}^{\infty} u_n + \mu \sum_{k=0}^{\infty} v_n
\] 
\subsubsection{Corollaire}
\[
	\begin{cases}
		\sum_{n\in \N}^{\infty} u_n &\in \R \\
		\sum_{n\in \N} u_n &\text{diverge}
	\end{cases} \implies \forall (\lambda, \mu) \in \K \times \K^\ast,\ \sum_{n\in \N} \lambda u_n + \mu v_n \text{diverge}
\] 
\subsection{Suites géométriques}
\begin{align*}
	|q| < 1 &\iff \sum_{\N} q^{\text{id}} \text{converge} \\
		&\implies \sum_{\N}^{\infty} q^{\text{id}} = \frac{1}{1-q}
\end{align*}
\subsection{Séries à termes positifs}
\[
	\sum_{n\in \N} u_n \implies \begin{cases}
		\sum_{\N}^{\infty} u \in \overline{\R} \\
		\text{suite des sommes partielles majorée} \iff \sum_{\N}^{\infty} u \in \R
	\end{cases}
\] 
\subsection{Principe de comparaison}
\newcommand{\convdiv}{{\text{converge} \atop \text{diverge}}}

\begin{align*}
	\begin{cases}
	\sum u, \sum v &\text{STP} \\
	\sum u &\convdiv \\
	u {\le \atop \ge} v &\text{ÀPCR}
\end{cases} \implies v &\convdiv
\end{align*}

\end{document}
