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\title{ESP Workshop, Worksheet \#5}
\date{\today}
\begin{document}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#5}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item  Eric is feeling really generous today! He tells you that these
  problems were on one of Professor Durbin's old M408D exams!

\begin{enumerate}
\item Determine the convergence or divergence of
\begin{displaymath}
\sum _{n=1} ^\infty \frac{1}{n^{\sqrt{2}}}
\end{displaymath}

\item Determine the convergence or divergence of
\begin{displaymath}
\sum _{n=1} ^\infty \frac{1}{5^n +3}
\end{displaymath}

\item Suppose the alternating series $\sum _{n=1} ^\infty (-1)^n a_n$
  converges to the sum $S$ and that $0< a_{n+1} \leq a_n$ for all $n$.
  What can you say about the difference between $S$ and $\sum _{n=1}
  ^N (-1)^n a_n$? (Eric says: ``Maybe it would help to draw a picture
  like Professor Durbin drew on the chalkboard yesterday...'')

\end{enumerate}

\item Use the limit comparison test to determine whether
\begin{displaymath}
\sum _{n=1}^\infty \frac{2n^2 +5}{2\sqrt{n + n^7}}
\end{displaymath}
converges or diverges.

\item It turns out that Eric actually needs a favor, and that's the
  only reason he was being nice! Eric wants to pass his Math Subject
  Test GRE so he teach calculus at UT, and he needs your help with
  some problems from his practice book!

\begin{enumerate}
\item Which of the following series converge, and which diverge?
\begin{displaymath}
(\textrm{i})\ \sum_{n=1} ^\infty \frac{\sin ^2 n}{n^3 +n} \qquad
(\textrm{ii})\ \sum_{n=0}^\infty \frac{1}{\sqrt{n+3}} \qquad
(\textrm{iii})\ \sum_{n=1} ^\infty \frac{\ln n}{n^2}
\end{displaymath}

\item Consider the sequence $\{a_n \}$ whose terms are given by the
  formula
\begin{displaymath}
a_n = \frac{(\cos n\pi)(\sin ^2 n)}{\sqrt{n}}
\end{displaymath}
for each integer $n\geq 1$. If this sequence converges, what is the
limit?

\item Which of the following series converge?
\begin{displaymath}
(\textrm{i})\ \sum_{n=1} ^\infty \frac{\cos ^4 (\arctan n)}{n\sqrt{n}} \qquad
(\textrm{ii})\ \sum_{n=2} ^\infty \frac{1}{n\ln n} \qquad
(\textrm{iii})\ \sum _{n=0}^\infty \frac{(n+1)^3}{5(n+2)(n+3)(n+4)}
\end{displaymath}

\item Which of the following statements are true?
\begin{enumerate}
\item If $a_n \geq 0$ for every $n$, then:
\begin{displaymath}
\sum_{n=1}^\infty a_n\ \textrm{converges} \Rightarrow \sum_{n=1}^\infty \sqrt{a_n}\ \textrm{converges}
\end{displaymath}

\item If $a_n \geq 0$ for every $n$, then:
\begin{displaymath}
\sum_{n=1}^\infty n a_n\ \textrm{converges} \Rightarrow \sum_{n=1}^\infty a_n\ \textrm{converges}
\end{displaymath}

\item If $a_n \geq 0$ and $a_{n+1} \leq a_n$ for every $n$, then:
\begin{displaymath}
\sum_{n=1}^\infty a_n^2 \ \textrm{converges} \Rightarrow \sum_{n=1}^\infty (-1)^n a_n\ \textrm{converges}
\end{displaymath}

\end{enumerate}

\item (Challenge.) If $|x| < 1$, then compute
\begin{displaymath}
\sum_{n=1} ^\infty nx^{2n}
\end{displaymath}

\end{enumerate}

\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.

\item (This one is for all you engineering majors...) Suppose you have
  a large supply of books, all the same size, and you stack them at
  the edge of a table, with each book extending farther beyond the
  edge of the table than the one beneath it.  Show that it is possible
  to do this so that the top book extends entirely beyond the table.
  In fact, show that the top book can extend any distance at all
  beyond the edge of the of the table if the stack is high enough (!).
  Use the following method of stacking: the top book extends half its
  length beyond the second book. The second book extends a quarter of
  its length beyond the third. The third extends one-sixth of its
  length beyond the fourth, and so on. HINT: consider centers of mass.

\end{enumerate}

\end{document}