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\title{ESP Workshop, Worksheet \#7}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#7}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}

\begin{enumerate}
\item Warm-up: Screech loves math, and he has figured out a way to
  send a secret calculus message from the past! Help me decode it!
  First, let $f(x) = \frac12 e^x$.
  \begin{enumerate}
  \item Sketch $y=f(x), y= -f(-x),$ and $y= f(x) - f(-x)$ on the same
    axes. Screech says, ``Sometimes, I Need Help!''
  \item What is $d/dx (f(x) - f(-x))$?
  \item Sketch $y=f(x), y=f(-x),$, and $y = f(x) + f(-x)$ on the same
    axes. Screech says, ``College is Obviously So Hard!''
  \item What is $d/dx (f(x) + f(-x))$?
  \item What do you notice about your answers for parts (b) and (d)? Does
    this phenomenon remind you of another famous pair of functions?
  \item What is Screech trying to tell you??
  \end{enumerate}
  
\item Suppose we have a series $\sum_{n=1}^\infty a_n$. What does it
  mean for that sum to be absolutely convergent? What about
  conditionally convergent? Give an example of an alternating series
  that is absolutely convergent and one that is conditionally (but not
  absolutely) convergent. Can you think of an example that absolutely
  converges but does not converge??

\vspace{0.3in}
  Okay, enough Mr. Nice Guy.  Let's see what you're really made of...

\item Find the limit of the sequences:
\begin{displaymath}
(a) \left \{ \sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \ldots \right \}
\qquad (b) \left \{ \sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots \right \}
\end{displaymath}

\item Find the values of $p$ for which the series is convergent.
\begin{displaymath}
(a) \sum_{n=2} ^\infty \frac{1}{n(\ln n)^p}
\qquad (b) \sum _{n=3}^\infty \frac{1}{n \ln n [\ln (\ln n) ]^p}
\qquad (c) \sum _{n=2}^\infty \frac{1}{n^p \ln n}
\end{displaymath}

\item Find all positive values of $x$ such that the series
\begin{displaymath}
\sum _{n=1} ^\infty x^{\ln n}
\end{displaymath}
converges.  Is this a power series?

\item For what values of $p$ is each series convergent?
\begin{displaymath}
(a) \sum_{n=1} ^\infty \frac{(-1)^{n-1}}{n^p}
\qquad (b) \sum _{n=1} ^\infty \frac{(-1)^n}{n+p}
\end{displaymath}

\item For which positive integers $k$ is the following series
  convergent?
\begin{displaymath}
\sum_{n=1} ^\infty \frac{(n!)^2}{(kn)!}
\end{displaymath}

\item (Challenge.) Find the sum of the series
\begin{displaymath}
\sum _{n=2} ^\infty \ln \left( 1-\frac{1}{n^2} \right)
\end{displaymath}
(Super challenge.) Can you tell me how this method of problem-solving
relates to the famous Riemann-zeta function (shown below)?
\begin{displaymath}
\zeta (s) = \sum_{n=1} ^\infty \frac{1}{n^s}
\end{displaymath}
Do you know the value (possibly $\infty$, hint hint) of this function
for any $s$? Later this semester, we will compute $\zeta (2)$. Do you
happen to know what it is?  If so, can you derive it??  By the way, if
you understand this function really well, you could win a million
dollars!! (See Eric for details.)

\end{enumerate}

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