\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{amssymb,amsmath}
\usepackage{amsbsy}
\usepackage{psfrag}
\usepackage{amscd}
\title{ESP Workshop, Worksheet \#4}
\date{\today}
\begin{document}
%\maketitle
\begin{center}
{\bf \Large ESP Workshop, Worksheet \#4}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
  
\item 
\begin{enumerate}
  
\item Define the two terms \textit{bounded} and \textit{monotonic} as
  they apply to sequences and say what you can about a sequence that
  is both bounded and monotonic. 
  
  For the following sequences, determine whether each is bounded,
  monotonic, and convergent or divergent.

\item $a_n = \frac{n!}{n^n}$
\item $a_n = \frac{1}{n}$

\end{enumerate}

\item Determine the convergence or divergence of $\sum _{n=1} ^\infty
  \frac{n^n}{n!}$. 

\item Evaluate:
\begin{enumerate}
\item $\sum_{i=1}^{100} 2$
\item $\sum_{i=1}^{100} \frac{1}{i+3} - \frac{1}{i+4}$
\item $\sum_{i=1}^\infty \frac{1}{i(i+1)}$
\end{enumerate}

\vspace{.2in}
\item In class we learned about  geometric series.  They are of the form
\[a +ar + ar^2+ar^3+\ldots\,\,=\sum_{i=0}^{\infty} ar^i\]
\begin{enumerate}
\item Let $s_n$ be the $n$-th partial sum of this series.
\[s_n=a+ar+ar^2+\ldots+ar^n\hspace{.25in}\textrm{(there are n+1 terms here)}\]
Calculate $s_n-rs_n$.
\item Using this, find a ``closed'' formula for $s_n$.
\item Compute $\lim_{n \to \infty}s_n$.  How does this limit depend on $r$?
\end{enumerate}

\vspace{.2in}
\item \begin{enumerate}
\item Converge or Diverge?
\[\frac{\sin(\theta)}{2}+\frac{\sin^2(\theta)}{4}+\frac{\sin^3(\theta)}{8}+\frac{\sin^4(\theta)}{16}+\ldots\]

If it converges can you tell what it converges to?
\item Converge or Diverge?
\[\sum_{n=1}^{\infty}\frac{\sin(4n)}{4^n}\]
If it converges can you tell what it converges to?
\end{enumerate}

\item If $a$ and $b$ are digits, show (using arguments involving
  geometric series like those we saw in class):
\begin{enumerate}
\item $0.\overline{a} = \frac{a}{9}$
\item $0.\overline{ab} = \frac{10a+b}{99}$
\item $.\overline{9} = 1$
\item Write $3.7\overline{2}$ as a fraction.
\end{enumerate}

\item 
\begin{enumerate}
\item Evaluate $\int _1 ^\infty \frac1x\ dx$, draw the graph of $y =
  1/x$ and indicate (graphically) what this integral is measuring.

\item For what values of $p$ does $\lim _{t\rightarrow \infty} t^{1-p}$ converge? Diverge?

\item Suppose $p < 1$. What is
\begin{displaymath}
\int _1 ^\infty \frac{1}{x^p}\ dx ?
\end{displaymath}

\item Now suppose $p > 1$. What is $\int _1 ^\infty x^{-p}\ dx$? What
  about
\begin{displaymath}
\int _1 ^\infty 2\pi x^{1-p}\ dx
\end{displaymath}
and what does it measure? (Hint: what's Eric's least-favorite hole in
putt-putt?) Draw a picture!

\item What does the integral test for convergence say?  Use it to
  decide for which $p$ the series $\sum _{n=1} ^\infty \frac{1}{n^p}$
  converges.

\end{enumerate}
  
\item Compute the area of the snowflake curve from the previous
  worksheet, and, if you haven't done so yet, show that it has
  infinitely long perimeter.

\item (Challenge.) Find the sum of the series
\begin{displaymath}
1+\frac12 +\frac13 +\frac14 +\frac16 +\frac18 +\frac19 +\frac{1}{12} +\cdots
\end{displaymath}
where the terms are the reciprocals of the positive integers whose
only prime factors are 2s and 3s.

\end{enumerate}

\end{document}