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\title{ESP Workshop, Worksheet \#8}
\date{\today}
\begin{document}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#8}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item You know a power series representation for
  $\displaystyle\frac{1}{1-x}$.  Recalling this, use term by term
  integration or differentiation, or direct substitution to find a
  power series representation for the following functions.  State the
  inverval of convergence for each power series.

\begin{tabular}{ll}
(a)\  $\displaystyle\frac{1}{(1-x)^2}$\hspace{1.5in}&(b)\  $\displaystyle\frac{1}{1+x}$\\
&\\
(c)\ $\displaystyle\frac{1}{1+x^2}$\hspace{1.5in}& (d)\ $\displaystyle \tan^{-1}(x)$\\
&\\
(e)\ $\displaystyle\ln\left(\frac{1+x}{1-x}\right)$\hspace{1.5in} & (f)\ $\displaystyle(\ln(1-x)-1)(1-x)$  
\end{tabular}

\vspace{.2in}
Hint for (f):  Take derivatives until you recognize what to do.

\vspace{.3in}
\item  Taylor series may be found for these functions by a long
method or a short method.  Describe both methods, then use the short
one.
$$
\textrm{(a)}\  f(x)={\sin x\over x}\hskip1in
\textrm{(b)}\  f(x)=x^3e^x\hskip1in
\textrm{(c)}\  f(x)=\ln(x^2+1)
$$

\item  Recall that a power series centered around $x=a$ has the general form
\begin{displaymath}
\sum_{n=0} ^\infty c_n (x-a)^n
\end{displaymath}
for some coefficients $c_n \in \mathbb{R}$. For each of the following
power series, find the center $a$, the coefficients $c_n$, the radius
of convergence $R$, and the interval of convergence. For the last
part, don't forget to check the endpoints!
\begin{displaymath}
\textrm{(a)}\ \sum_{n=0}^\infty{(x-2)^n\over n!} \qquad
\textrm{(b)}\ \sum_{n=0}^\infty{(x+3)^n\over5^n}
\end{displaymath}
\begin{displaymath}
\textrm{(c)}\ \sum_{n=0}^\infty(-1)^n{(x+4)^n\over n+2} \qquad
\textrm{(d)}\ \sum_{n=0}^\infty n!(x-1)^n
\end{displaymath}

\item Let
\begin{displaymath}
f(x) = \sum_{n=1} ^\infty \frac{x^n}{n^2}
\end{displaymath}
Find the intervals of convergence for $f, f',$ and $f''$.

\item
\begin{enumerate}
\item Starting with the geometric series $\frac{1}{1-x} = \sum_{n=0}
  ^\infty x^n$, find the sum of the series
\begin{displaymath}
\sum_{n=1} ^\infty nx^{n-1} \quad |x| < 1
\end{displaymath}

\item Find the sum of each of the following series.
\begin{displaymath}
(i)\ \sum_{n=1} ^\infty nx^n,\ |x| < 1 
\qquad (ii)\ \sum_{n=1} ^\infty \frac{n}{2^n}
\end{displaymath}

\item Find the sum of each of the following series.
\begin{displaymath}
(i)\ \sum_{n=2} ^\infty n(n-1) x^n,\ |x| < 1
\qquad (ii)\ \sum_{n=2} ^\infty \frac{n^2 -n}{2^n}
\qquad (iii)\ \sum_{n=1}^\infty \frac{n^2}{2^n}
\end{displaymath}


\end{enumerate}

\vspace{.3in}
\item In class the other day, Professor Durbin observed that the
  Taylor series centered around $x=0$ of $f(x) = \sin x$ (an odd
  function) had no terms with even powers of $x$. Can you prove that
  this is true for odd functions in general?  That is, can you show
  that if $f$ is an odd function then the Taylor series for $f$ at
  zero will contain only terms with odd powers?

\item
\begin{enumerate}
\item Show that the function defined by
\begin{displaymath}
f(x) = \left\{ \begin{array}{lc}
e^{-1/x^2} & \textrm{if}\ x\neq 0 \\
0 & \textrm{if}\ x=0
\end{array} \right .
\end{displaymath} 
is not equal to its Maclaurin series.
\item Graph the functions $e^x, -1/x^2,$ and $e^{-1/x^2}$. Use the
  third graph to comment on the behavior near the origin of $f(x)$
  from part (a).
\end{enumerate}

\item (Challenge.) Find the sum of the series
\begin{displaymath}
\sum _{n=2} ^\infty \ln \left( 1-\frac{1}{n^2} \right)
\end{displaymath}


\end{enumerate}
\end{document}