Math. 401, Sec. 500 Spring, 2005
Homework 9, due April 1
1. [Logan, p. 195, Ex. 3.2–3]
(a) Let f(x)=x
2on −π≤x≤πand f(x+2π)=f(x) for all x. Show that
f(x)=π
2
3+4
∞
X
n=1
(−1)ncos(nx)
n2.
(b) Use this series to prove that π2
12 =1−1
4+1
9−1
16 +···.
2. [Logan, p. 195, Ex. 3.6] Find the Fourier series for the periodic function defined by
f(x)=(0 for −π<x<0,
1 for 0 ≤x<π.
To what value does the series converge at x=0?
3. [Schaum’s, p. 46, Ex. 2.34–5(b)] If
f(x)=(−xwhen −4≤x≤0,
xwhen 0 ≤x≤4,
and fis 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 discon-
tinuities of fare located and to what value the series converges at each discontinuity.
4. [Schaum’s, p. 46, Ex. 2.34–5(c)] If f(x)=4xfor 0 <x<10 and fis periodic with
period 10 (note: not 20), graph the function and find its Fourier series. Also, tell
where the discontinuities of fare located and to what value the series converges at
each discontinuity.
5. [Schaum’s, p. 46, Ex. 2.38 ]
(a) Expand f(x)=cosx,0<x<π, in a (“full”) Fourier series.
(b) Expand f(x)=cosx,0<x<π, in a Fourier cosine series.
(c) 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.
6. [Schaum’s, p. 46, Ex. 2.40] Show that for 0 ≤x≤π,
(a) x(π−x)=π
2
6−cos 2x
12+cos 4x
22+cos 6x
32+···
.
(b) x(π−x)= 8
πsin x
13+sin 3x
33+sin 5x
53+···
.
