\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Define the function $f$ by
$f(x)=\sum\limits_{n=0}^\infty (-1)^n
{{x^{2n}}\over{n!}}=1-x^2+{{x^4}\over2}-\ldots$
(remember that $0!=1$). 

\medskip

\noindent a) Determine the interval of convergence; this is the domain of $f$. 

\medskip

\noindent b) Write out several terms of the series and verify that
$f'(x)=-2xf(x)$ for all $x$ in the interior of the interval of
convergence. 

\medskip

\noindent c) Show that $y=f(x)$ is a solution of the initial value problem
$y'=-2xy,\ \  y(0)=1.$

\medskip

\noindent d) Solve this initial value problem and get a formula for
$f(x)$ in terms of functions found on your calculator.

%Possible continuation of this problem, or use as an additional
%problem. 

\medskip

\noindent e) Use the formula discovered for $f(x)$ and graph both $f$
and the partial sum $s_6(x)=1-x^2+{{x^4}\over2}-{{x^6}\over6}$ in a
window where $0\le x\le 1.2$.  Then use the alternating series error
formula to obtain an upper bound for the error in the approximation
$f(x)\approx s_6(x)$ when $0\le x\le 1.2$. Your answer should be a
single number that applies to all $x$ values in the range $0 \leq x
\leq 1.2$, and it should be consistent with the graphs you have drawn.

\vfil\eject\end

