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\noindent {\bf Problem statement} Define $f(x)$ with the sum
$f(x)=\sum\limits_{n=0}^\infty {{2^n\cos(nx)}\over{n!}}.$ This series
is \underbar{not} a power series.  To the right is a graph of the
partial sum $s_{100}(x)=\sum\limits_{n=0}^{100}
{{2^n\cos(nx)}\over{n!}}$ for $0\le x\le 20$.}

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\noindent a) Verify that the series defining $f(x)$ converges for all $x$.

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\noindent b) Is the apparent periodicity of the function $f(x)$
actually correct? If yes, explain why. Your explanation should
include use of the term ``periodic function'' and explain why the
defining condition is or is not satisfied. 

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\noindent c) Verify that the actual graph of the function is always
within .01 of the graph shown. That is, if $x$ is any real number,
then $|f(x)-s_{100}(x)|<.01$.  %$e^{2\cos x}\cos(2\sin x)$

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\noindent {\sevenbf Possibly useful numbers} $\scriptstyle 2^{100}\approx
1.27\cdot 10^{30}\hbox{\sevenrm and }2^{101}\approx 1.54\cdot
10^{30}$. $\hbox{\sevenrm Also, }\scriptstyle 100!\,\approx 9.33\cdot 10^{157}\hbox{\sevenrm and }
101!\,\approx 9.43\cdot 10^{159}.$











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