\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} One of the Bessel functions used to
describe the vibration of a circular plate is defined by this infinite
series:
$\displaystyle J(x)=\sum_{n=0}^\infty {{(-1)^nx^{2n}}\over{2^{2n}(n!)^2}}\,.$

\medskip

\noindent a) Show that this series converges absolutely for all values of $x$.

\medskip


\noindent b) Explain briefly why the result of a) implies that the series
converges for all $x$.

\medskip

\noindent c) Here are individual terms of the series for two values of $x$
and for some values of $n$.

\smallskip

\def\lct#1#2{\vhr\nhr\vhr&& #1
&&\hfil #2 \hfil&\cr}
\def\nhr{\noalign{\hrule}}
\def\vhr{\omit&height 2pt&&&&\cr}
\def\tskip{\noalign{\smallskip}}
\def\nhrt{\noalign{\hrule width .02in\smallskip}}

\def\vhr{\omit&height 3pt&&&&&&&&&&&&&&\cr}
\def\vvhr{\omit&height 4pt&&&&&&&&&&&&&&\cr}
\def\vvvhr{\omit&height 5pt&&&&&&&&&&&&&&\cr}
\medskip
\centerline{
\vbox{\offinterlineskip
\halign{
\strut#&\vrule width .0085in # &
# & \vrule # &
# & \vrule # &
# & \vrule # &
# & \vrule # &
# & \vrule # &
# & \vrule # &
# & \vrule # &
& # \vrule width .0085in &# \cr
\nhr\nhr\vvvhr
&&
$\ \ \displaystyle{{(-1)^nx^{2n}}\over{2^{2n}(n!)^2}}$
&&
\ \hfil\ $n=0$\hfil \
&&
\ \hfil\ $n=1$\hfil\
&&
\ \hfil\ $n=2$\hfil \
&&
\ \hfil\ $n=3$\hfil \
&&
\ \hfil\ $n=4$\hfil  \
&&
\ \hfil\ $n=5$\hfil \
&\cr
\vhr\nhr\vvhr
%\noalign{\hrule}
&&
$\hfil\ \ x=1$\hfil
&&
\hfil\ $1$\hfil
&&
\hfil\ $-{1\over 4}$\hfil
&&
\hfil\ ${1\over {64}}$\hfil
&&
\hfil\ $-{1\over {2,304}}$\hfil
&&
\hfil\ ${1\over{147,456}}$\hfil
&&
\hfil\ $-{1\over{14,745,600}}$\hfil
&\cr
\vvhr\nhr\vvhr
&&
\hfil$\ \ x=4$\hfil
&&
\hfil\ $1$\hfil
&&
\hfil\ $-4$\hfil
&&
\hfil\ $4$\hfil
&&
\hfil\ $-{{16}\over 9}$\hfil
&&
\hfil\ ${4\over 9}$\hfil
&&
\hfil\ $-{{16}\over{225}}$\hfil
&\cr
\vvvhr\nhr\nhr
}}}
\medskip

\noindent Use entries of this table and facts about the series to
explain why $J(1)$ must be positive and $J(4)$ must be negative. 

\medskip

\noindent {\bf Hint} Select an $N$ for each $x$ and split the sum:
$\sum\limits_{n=0}^\infty=\sum\limits_{n=0}^N+\sum\limits_{n=N+1}^\infty$.
Evaluate the finite sum explicitly and estimate the infinite tail
$\sum\limits_{n=N+1}^\infty$. 










\vfil\eject\end

