\documentclass[letterpaper,12pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage[letterpaper,margin=1in,includehead=true]{geometry}
\usepackage{comment}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{enumerate}
\pagestyle{fancy}
\usepackage{color}
\usepackage{wrapfig}
%\usepackage{enumitem}
%\usepackage{tikz, tikz-3dplot, pgfplots}
%\usetikzlibrary{calc}
%\usepackage{titling}
%\setlength{\droptitle}{-1.5in}

%opening
\title{\vspace*{-1.5in}\textbf{\S8.4: Absolute Convergence and Ratio Test}\\\normalsize (Thanks to Faan Tone Liu)}
\date{}
\author{}

\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{March 7, 2018}
\rhead{\S8.4: Abs. Conv., Ratio Test}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item If the series $\sum |a_n|$ converges, then the original series $\sum a_n$ converges: \vskip1cm
In this case, we say the series $\sum a_n$ is \textbf{absolutely convergent} or \textbf{converges absolutely}. This is a new way to prove series convergence!

\item Series that are convergent but are not absolutely convergent are called \textbf{conditionally convergent}. One example is\vskip1.5cm

\item The \textbf{Ratio Test} is another tool we can use. Calculate $\displaystyle \lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right| = L$. 
\begin{enumerate}[(i)]
\item If $L < 1$, then $\sum a_n$ converges absolutely (and thus converges)
\item If $L > 1$, including $+\infty$, then $\sum a_n$ diverges
\item If $L=1$, the ratio test tells you nothing. Try something else. 
\end{enumerate}

\item Helpful graphic:
\begin{center}
\begin{tabular}[t]{l|c|c}
&$\sum a_n$ converges & $\sum a_n$ diverges\\[1mm]\hline&&\\[-6pt]
$\sum |a_n|$ converges &&\\[6pt]\hline &&\\[-6pt]
$\sum |a_n|$ diverges &&\\[6pt]
\end{tabular}
\end{center}
\end{itemize}
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Does $\displaystyle \sum_{n=1}^\infty \frac{\sin(n)}{n^2}$ converge or diverge?\vfill
\item Does $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n^2}$ converge or diverge?\vfill
\newpage
\item Does $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n+1}}$ converge absolutely, converge conditionally, or diverge?\vfill
\item Does $\displaystyle \sum_{n=1}^\infty \frac{n^2}{2^n}$ converge absolutely, converge conditionally, or diverge?\vfill
\item Does $\displaystyle \sum_{n=1}^\infty \frac{(-1)^nn!}{e^n}$ converge absolutely, converge conditionally, or diverge?\vfill
\item Does $\displaystyle \sum_{n=1}^\infty \frac{n^n}{(2n)!}$ converge absolutely, converge conditionally, or diverge?\vfill

\end{enumerate}
\end{document}
