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\title{\vspace*{-1.5in}\textbf{\S8.1: Sequences}\\\normalsize(Created by Faan Tone Liu)}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{Feb. 20, 2018}
\chead{\S8.1: Sequences}
\vspace{-.5in}
\noindent
\textbf{Key Points:}\vskip12pt
\begin{itemize}
\item Think of a sequence as a comma-separated list: \[a_1, a_2, a_3, \ldots, a_n,\ldots.\]
\item A sequence is a function whose domain is the positive integers. You can graph a sequence of real numbers. 
\item We are often interested in the end behavior of a sequence, $\displaystyle \lim_{n\to \infty}a_n$. Hint: use the ``connect the dots'' function defined on $\mathbb{R}$ (i.e. $f(n) = a_n$). In particular, $$\lim_{n \to \infty} a_n = \lim_{x \to \infty} f(x),$$ and we can use Calc I tools like L'H\^{o}pital's Rule. 
\item Some neat tools:
\begin{itemize}
\item Squeeze Law (Sandwhich Theorem):\vskip1cm
\item Showing an alternating sequence converges:\vskip1cm
\item Showing an alternating sequence diverges:\vskip1cm
\item Three ways to show a sequence is decreasing
\begin{enumerate}
\item\vskip12pt
\item\vskip1cm
\item\vskip1cm
\end{enumerate}\vskip1cm
\item Note: Bounded Monotonic (i.e. increasing or decreasing) sequences must converge. 
\end{itemize}
\end{itemize}
\newpage
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Consider the sequence $\displaystyle 1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \ldots, a_n, \ldots.$ Find a formula for $a_n$.\vskip1.2in
\item Consider the sequence $\displaystyle \frac{1}{3}, \frac{1}{6}, \frac{1}{11}, \frac{1}{18}, \ldots, a_n, \ldots.$ Find $a_n$.\vskip1.2in
\item Consider the sequence $\displaystyle \frac{2}{3}, \frac{4}{9}, \frac{6}{27}, \frac{8}{81}, \ldots, a_n, \ldots.$ Find $a_n$.\vskip1.2in
\item Consider the sequence $\displaystyle -\frac{5}{2}, \frac{8}{4}, -\frac{11}{8}, \frac{14}{16}, \ldots, a_n, \ldots.$ Find $a_n$.\vskip1.2in
\item Consider the sequence $\displaystyle 7, -\frac{9}{2}, \frac{11}{6}, -\frac{13}{24}, \ldots, a_n, \ldots.$ Find $a_n$.\vskip1.2in
\newpage
\item Suppose $\displaystyle a_n = \frac{\cos n}{n^2}$. Find $\displaystyle \lim_{n\to \infty}a_n$.\vskip1.2in
\item Suppose $\displaystyle a_n = \frac{(-1)^n\ln{n}}{n}$. Find $\displaystyle \lim_{n\to \infty}a_n$.\vskip1.2in
\item Suppose $\displaystyle a_n = \frac{(-1)^n\left(n^3 + 3\right)}{2n^3-1}$. Find $\displaystyle \lim_{n\to \infty}a_n$.\vskip1.2in
\item Suppose $\displaystyle a_n = \frac{\sqrt{3n^2 + 4}}{n-1}$. Find $\displaystyle \lim_{n\to \infty}a_n$.\vskip1.2in
\item Suppose $\displaystyle a_n = \left(1+\frac{1}{n}\right)^n$. Find $\displaystyle \lim_{n\to \infty}a_n$.\vskip1.2in
\newpage
\item Show $\displaystyle a_n = \frac{3^{n+2}}{5^n}$ is decreasing.\vskip1.2in
\item Show $\displaystyle a_n = \frac{n}{n+1}$ is increasing.\vskip1.2in
\item Show $\displaystyle a_n = \frac{n}{e^n}$ is decreasing.\vskip1.2in
\item Find a formula for $a_n$ if $a_1 = 2$ and $a_{n+1} = a_n + 5$. \vskip1.2in
\item Find a formula for $a_n$ if $a_1 = 4$ and $a_{n+1} = 3\cdot a_n$. \vskip1.2in
\end{enumerate}
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