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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 15\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 15} 


\bigskip 

\item{1.}  Let $\{a_n\}_{n=0}^\infty$ be a sequence such that
$\sum_{n=0}^\infty a_n9^n$ converges and that
$\sum_{n=0}^\infty (-1)^na_n12^n$ diverges.  What if anything can be
said about 
$$
\eqalign{
\pt a &\hbox{the convergence of}\sum_{n=0}^\infty a_n7^n\cr
\pt b &\hbox{the absolute convergence of}\sum_{n=0}^\infty a_n(-7)^n\cr
\pt c &\hbox{the absolute convergence of}\sum_{n=0}^\infty a_n9^n\cr
\pt d &\hbox{the convergence of}\sum_{n=0}^\infty (-1)^na_n9^n\cr}\hskip.5in
\eqalign{
\pt e &\hbox{the convergence of}\sum_{n=0}^\infty a_n10^n\cr
\pt f &\hbox{the convergence of}\sum_{n=0}^\infty a_n(-15)^n\cr
\pt g &\hbox{the convergence of}\sum_{n=0}^\infty a_n15^n\cr
\pt h &\hbox{the convergence of}\sum_{n=500}^\infty a_n15^n\cr}\hskip.9in
$$

\item{2.}  Sketch and label on a number line the regions where you
know the series in 
\item{}  Problem 1 converges and where it diverges.  What can be said
about  
$$
\lim_{n\to\infty}{a_{n+1}\over a_n}?
$$  

\item{3.}  Let $f(x)$ be a {\it smooth} function, that is, the
derivatives $f^{(n)}(x)$ exists for all 
\item{}  $n=1,\,2,\,3,\,\ldots$ The {\it Taylor series for $f(x)$
about $x=a$} is defined to be 
$$
T_a(x)=\sum_{n=0}^\infty{f^{(n)}(a)\over n!}(x-a)^n.
$$
Find the Taylor series about $x=a$ for the following:
$$
\pt a f(x)=x^2 \hskip1in 
\pt b f(x)=x \hskip1in 
\pt c f(x)=2^x
$$


\eject
\item{4.}  Consider the function $f(x)=e^{-1/x^2}$.

\medskip
\itemitem{a)}  Graph $f(x)$ {\bf without the aid of a calculator}.

\medskip
\itemitem{b)}  Notice that $f(0)$ is not defined.  (You did indicate
that in your graph didn't you?)  Define $f(0)$ to be a value so that
$f(x)$ is continuous at $x=0$.  

\medskip
\itemitem{}  You may either take my word for it or prove that $f(x)$
is now infinitely differentiable at $x=0$.  That is, not only have you
made $f(x)$ continuous at $x=0$, but in fact, all of its derivatives
$f^{(n)}(x)$ exist there.

\medskip
\itemitem{c)}  Find the Taylor series for $f(x)$ centered at $x=0$.
Does this equal $f(x)$?

\bigskip
\item{5.}  Important in the theory of probability is the {\it normal density
function} 
$$
n(x)={1\over\sqrt{2\pi}}e^{-{1\over2}x^2}.
$$

\itemitem{a)}  Graph $n(x)$.

\medskip
\itemitem{b)}  Various probabilites can be computed as the area under
certain sections of this curve.  Typically, for various values of $x$,
we may want to compute 
$$
N(x)={1\over\sqrt{2\pi}}\int_{-\infty}^xe^{-{1\over2}y^2}\,dy
$$
which is called the {\it normal distribution function}.  What makes
computing this integral difficult?

\medskip
\itemitem{c)}  Find the Taylor series centered at $x=0$ for the normal
distribution $N(x)$.  Why might this be useful?






\bye