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\title{ESP Workshop, Worksheet \#6}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#6}

{\bf \large Tuesday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item Identify in each problem below which test you would use to determine convergence. Do not do the work, just identify the most effective test.

\begin{displaymath}
\textrm{a)\ } \sum_{n=2} ^\infty \frac{1}{n(\ln n)^2} \qquad
\textrm{b)\ } \sum_{n=1} ^\infty \frac{(n+1)^2}{n(n+2)} \qquad
\textrm{c)\ } \sum_{n=1} ^\infty \frac{1}{(2n)!}
\end{displaymath}

\begin{displaymath}
\textrm{d)\ } \sum_{n=1} ^\infty \frac{n3^n}{4^{n-1}} \qquad
\textrm{e)\ } \sum_{n=1} ^\infty \frac{n^n}{n!} \qquad
\textrm{f)\ } \sum_{n=1} ^\infty \frac{5}{\sqrt{n^2 - 1}}
\end{displaymath}

\vspace{.2in}
\item Let $x$ be a real number. Use the ratio test to determine the
  values of $x$ for which the series $\sum_{n=1}^\infty
  \frac{x^n}{n^3}$ converges. Try the same for $\sum _{n=1}^\infty
  \frac{x^n}{n!}$. (For future reference, this set of values is
  commonly called the ``radius of convergence''.)

  
\vspace{.2in}
\item Test each series for convergence or divergence. For convergent
  alternating series, classify the convergence as absolute or
  conditional.

\begin{displaymath}
\textrm{a)\ } \sum_{n=1} ^\infty \frac{(-1)^{n+1} (n+1)}{n^2 +n+1} \qquad
\textrm{b)\ } \sum_{n=1} ^\infty \left( \frac{2n}{1+8n}\right)^n \qquad
\textrm{c)\ } \sum_{n=1} ^\infty \frac{(-2)^{n+1}}{3^n}
\end{displaymath}

\begin{displaymath}
\textrm{d)\ } \sum_{n=1} ^\infty \ln \left( \frac{2n-1}{n+4}\right) \qquad
\textrm{e)\ } \sum_{k=1} ^\infty \frac{2^k k!}{(k+2)!} \qquad
\textrm{f)\ } \sum_{n=1} ^\infty e^{-n^2}
\end{displaymath}

\begin{displaymath}
\textrm{g)\ } \sum_{n=1} ^\infty \sin ^3 \left( \frac{1}{n}\right) \qquad
\textrm{h)\ } \sum_{n=1} ^\infty \frac{\ln n}{n^3 -1} \qquad
\textrm{i)\ } \sum_{n=1} ^\infty \frac{n3^n}{4^{n-1}}
\end{displaymath}

(Hint for part (g): First make a graphical argument to show that for
all positive $x$, $x \geq \sin x$. What is the derivative of $\sin x$
at $x=0$?)

\newpage
\item Define a function $f(x)$ by the following series:
\begin{displaymath}
f(x) = \sum_{n=0} ^\infty x^n
\end{displaymath}
\begin{enumerate}
\item This is a power series. What is the ``center'' of this power
  series? (That is, what is the center of the interval of
  convergence?)
  
\item What is the interval of convergence of this function, i.e.  for
  what values of $x\in \mathbb{R}$ does the series converge?
  
\item Find a simpler expression for $f(x)$. Find its derivative, and
  then graph $y = f(x)$.
  
\item What is the value of $f(-1)$? Use both your simpler expression
  from part (b) and the original series defintion to evaluate it. What
  happened??

\end{enumerate}

\vspace{.2in}
\item  For each power series, find the radius of convergence and determine 
if it converges at each of the endpoints of the interval of convergence.

a) $\displaystyle \sum_{n=1}^{\infty} \frac{(3x-2)^n}{\sqrt{n}}$
\hspace{.4in}
b) $\displaystyle \sum_{n=1}^{\infty} x^n n^n$
\hspace{.4in}
c) $\displaystyle \sum_{n=1}^{\infty} \frac{6^n}{n^2} x^n$
\hspace{.4in}
d) $\displaystyle \sum_{n=1}^{\infty} \frac{(2x)^n}{n^n}$

\vspace{.2in}
\item Geometric and power series are often confused.  Power series are
  of the form $\displaystyle \sum c_n x^n$, where the $c_n$ are
  possibly different constants, $x$ an unknown constant.\footnote{We
    sometimes call $x$ a variable, especially when we're thinking of
    the power series as defining a function $f(x)$. However, for each
    value of $x$, we may think of $x$ as a constant in the power
    series since it does not depend on the index $n$.}  Geometric
  series are of the form $\displaystyle \sum a r^n$ where $r$ and $a$
  are constants.  For each series below, determine if it is geometric,
  power, both or neither.

\vspace{.2in}
a) $1 + \frac{x}{2} + \frac{x^2}{4} + \frac{x^3}{8} + ...$ \hspace{.3in}
b) $1 + 1.1 + 1.21 + 1.331 + 1.4641 + ...$ \hspace{.2in}
c) $(\frac{1}{3})^2 + (\frac{1}{3})^4 + (\frac{1}{3})^6 + ...$

\vspace{.2in}
d) $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...$ 
\hspace{.2in}
e) $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$ \hspace{.4in}
f) $\frac{\ln x}{2} + \frac{\ln x}{3}+ \frac{\ln x}{4} + \frac{\ln x}{5} 
+ ...$

\vspace{.2in}
\item We say that a plane figure has ``$n$-degree rotational
  symmetry'' if it looks the same after rotating it by $n$ degrees
  around some point.  For example, a square has 90-degree rotational
  symmetry. Now you're told that a certain plane figure has 19-degree
  rotational symmetry.  Prove that it also has 1-degree rotational
  symmetry.  Can you sketch such a figure?

\end{enumerate}

\end{document}