Solutions to Homework 1
Due Friday, June 25, 2004.
Chapter 1.1.
Problem 1 solution. If is an even linear function then must be constant. If z (w, 1) = z (w, 1) = 0 then (w) (1) = (w) (1) = 0. The
rst part of the exercise explains that if is identically ze
Solutions to Homework 2
Due Tuesday, July 6, 2004.
Chapter 2.3
Problem 3 solution. If the series for ln(1+z ) and ln(1z ) both converge,
1+
then we can nd the series for ln 1z by term-by-term subtraction of
z
the two series:
ln
1+z
1z
= ln(1 + z ) ln(1 z
Solutions to Homework 3
Due Tuesday, July 13, 2004.
Chapter 3.1
Solution to problem 1. Less than 0.001: k = 11 works. Less than
0.000001: k = 15 works. (Note. I dont know why the author wants
to use Stirlings estimate, since one can do the exercise and ge
Solutions to Homework 4
Due Tuesday, July 20, 2004.
Chapter 3.3) 1, 2, 7, 9, 17, 19.
Solution to problem 1. Cauchys proof requires that i) f is dierentiable
at every point in [a, b] ii) f is bounded on [a, b] and iii) for any > 0 there
is a > 0 such that
Solutions to Homework 5
Due Tuesday, July 27, 2004.
Chapter 3.4) 2, 3, 5, 10, 11, 14, 15, 22, 23, 26.
Solution to problem 2. Here is a graph:
0.1
0.08
0.06
0.04
0.02
-0.1
-0.05
0.05
0.1
It looks like = 0.1 might work, but to be on the safe side we may as
Solutions to Homework 7
Due Wednesday, August 4, 2004.
Chapter 4.1) 3, 4, 9, 20, 27, 30. Chapter 4.2) 4, 9, 10, 11, 12.
Chapter 4.1.
Solution to problem 3. The sum has the form
a1 a2 + a3
with ak = 1/k . Since the ak are positive and decreasing, the seri
Solutions to Homework 8
Due Friday, August 6, 2004.
Chapter 4.3) 1f, 7f.
n+1
Solution to problem 1f. Its easy to see that | aan | =
a rational function of n.
(2n+1)2
(n+1)2
which is
Solution to problem 7f. From the above we see the radius of convergence i
Solutions to Homework 9
Due Friday, August 13, 2004.
Chapter
Chapter
Chapter
Chapter
4.4)
4.5)
5.1)
5.2)
3
9, 12
2, 3, 10, 11
4, 5.
Chapter 4.4.
Solution to problem 3. Let f (x) = x(ln 1 )3/2 . It is positive, decreasing,
x
and tends to 0 at innity and th
Math 650 Summer 2004 Test 1. Wednesday, July 7, 2004
Name:
1. Fouriers series for a function equal to 1 on the interval (1, 1) is given by
f (x) =
4
x 1
3x 1
5x 1
7x
cos
cos
+ cos
cos
+ .
2
3
2
5
2
7
2
a) Find f (3/2).
b) Find f (3/2).
Solution. Keeping
Topics to Review for Test 2
Chapter 3.1. Newton-Raphson method. Conditions that guarantee convergence.
3.2. Dierentiability. Why derivatives of innite sums may not be the
2
sum of the derivatives. Strange examples: x4/3 sin(1/x), e1/x .
3.3. Mean Value Th
Math 650 Summer 2004 Test 2 Solutions. Friday, August 6, 2004.
Name:
1. Use the Mean Value Theorem to prove that if f is continuous on [a, b],
dierentiable on (a, b) and f (x) > 0 for all x in (a, b), then f is strictly
monotonically increasing on [a, b].