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 \centerline{\large Math.\ 401, Sec.\ 500\hfill    Spring, 2005}
 \bigskip
\centerline{\bfseries\large Homework 9, due April 1}
\bigskip
                                          
                                      
\begin{enumerate}

\item %\vskip-\baselineskip
{\it [Logan, p.\ 195,  Ex.\ 3.2--3]\/} \
\begin{enumerate}
\item
Let $f(x) = x^2$ on $-\pi\le x\le\pi$ and
$f(x+2\pi) = f(x)$ for all~$x$.
Show that
$$f(x) = {\pi^2\over 3} + 4 \sum_{n=1}^\infty (-1)^n 
{\cos (nx) \over n^2} \,.$$


\item %{\it [Logan, p\. 195,  Ex\. 3.3]\/} \
Use this series to prove that \
$\displaystyle {\pi^2\over12} = 1 -\frac14 + \frac19 -\frac1{16} 
+ \cdots$.
\end{enumerate}

\item {\it [Logan, p.\ 195,  Ex.\ 3.6]\/} \
Find the Fourier series for the periodic function defined by
$$ f(x) = \left\{\begin{array}{ll}
 0 & \mbox{for $-\pi < x < 0$,} 
\\
1 & \mbox{for $0 \le x < \pi$.} 
\end{array}\right.$$ 
To what value does the series converge at $x=0$?

\item {\it [Schaum's, p.\ 46, Ex.\ 2.34--5(b)]\/} \ 
If 
$$ f(x) = \left\{ \begin{array}{ll}
 -x & \mbox{when $-4\le x \le 0$}, \\
   x & \mbox{when $0\le x \le 4$}, 
\end{array}\right.
$$
and $f$ is periodic with period $8$, graph the function and find 
its Fourier series (using properties of even or odd functions 
whenever applicable). 
Also, tell where the discontinuities of $f$ are located and to 
what value the series converges at each discontinuity.


\item {\it [Schaum's, p.\ 46, Ex.\ 2.34--5(c)]\/} \ 
If 
$f(x) = 4x $ for $0 < x < 10$
and $f$ is periodic with period $10$ (note: {\bf not} $20$),
     graph the function and find 
its Fourier series. 
Also, tell where the discontinuities of $f$ are located and to 
what value the series converges at each discontinuity.




\item {\it [Schaum's, p.\ 46, Ex.\ 2.38\/]} \
\begin{enumerate}
\item Expand $f(x)=\cos x$, $0<x<\pi$, in a 
(``full'') Fourier series.
\item Expand $f(x)=\cos x$, $0<x<\pi$, in a 
 Fourier cosine series.
\item Compare these results with the Fourier sine series 
you found for this function last week (Exercise~6 of Homework~8).
Explain the differences (if any) among them.
\end{enumerate}
 
\item {\it [Schaum's, p.\ 46, Ex.\ 2.40]\/} \ 
Show that for $0\le x\le \pi$,
\begin{enumerate}
\item $\displaystyle x(\pi-x) = \frac{\pi^2}6 - \left( {\cos 
2x\over 1^2} + 
{\cos 4x\over 2^2} + {\cos 6x\over 3^2} + \cdots \right)$.
\item \ $\displaystyle x(\pi-x) = \frac8{\pi} \left( {\sin x\over 
1^3} + 
{\sin 3x\over 3^3} + {\sin 5x\over 5^3} + \cdots \right)$.
\end{enumerate}

\item {\it [Schaum's, p.\ 46, Ex.\ 2.41)]\/} \ 
Use the results of Exercise~6 to show
\begin{enumerate}
\item \ $\displaystyle \sum_{n=1}^\infty \frac1{n^2} = 
\frac{\pi^2}6\,$.

\item \ $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} = 
\frac{\pi^2}{12}\,$.

\item \ $\displaystyle \sum_{n=1}^\infty 
\frac{(-1)^{n-1}}{(2n-1)^3} = 
\frac{\pi^3}{32}\,$.
\end{enumerate}

\end{enumerate}
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