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Math 436

October 9, 2000

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{\bf MIDTERM}
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On questions with yes or no answers, you must give a reason.  Show all
your work.  If you use a theorem, state that you are using it.
 There is a table of Fourier series at the end of the 
exam, which you may find useful.  
If you find you are doing a horrendous calculation, you are
making a mistake or at least making the problem unnecessarily
difficult.

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1. (10 points) Find the Fourier series of  $\cos 7136481x\/$
on the interval  $-\pi<x<\pi\/$.  (Hint: Think!)

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2. (10 points) Does the Fourier sine series of  $x\/$
converge to  $x\/$
on  $(0,\pi)\/$?  If so, does it converge
uniformly?  

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3. (15 points) Find:

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a)
$
\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\,.
$


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b)
$
\sum_{n=1}^\infty \frac 1{n^2}\,.
$

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4. (15 points)  Find all separated solutions of:
\begin{eqnarray*}
& u_t= u_{zz}\,, &
\!\!\!\!\qquad -\pi<z<\pi,\ t>0,  
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\\
& u_z(-\pi,t)&\!\!\!\!=0=u_z(\pi,t), 
\end{eqnarray*}


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