Math 4032, Test # 1. Spring 2006
Name:
1[10P]) True or false:
a) Let f : (a, b) R be continuous, then f is dierentiable. (F) [Take f (x) = |x|)
b) The function f (x) =
x sin(1/x) , x = 0
is continuous but not dierentiable. (T)
0
, x=0
Answer 3 of the foll
An idea how to solve some of the problems
5.2-2. (a) Does not converge: By multiplying across we get
1/2
2k
2k 2 k 2 1/2 k 2 1/2
1
k
2k 2
Hence
2k
1/2
.
1
k
2k 2
1
As the series k diverges the same must hold for the original series.
k =1
(b) Converges: W
Solution to the homework due 3-31-06
5.5-5: (a) The sequence
we have
kx
k =1 e
converges uniformly on [1, ). For that note that on this interval
ekx ek = (1/e)k
and the series (1/e)k converges. The claim follows then by the Weierstrass M -test.
k =1
kx
(b
Math 4032, Solution to homework, March 24
5.4-3 Let V be a normed linear space.
(a) If T V , prove |T (v )| T
v for all v V .
(b) If T V , prove T = 0, if and only if T = 0.
Solution: Recall denition 5.4.2 that
T = inf cfw_K | |T (v )| K v v V .
For simp
Things to prove today from 3, 4 and 5.1
Theorem 3.4.2 p. 87.
Example 4.1.1.
Theorem 4.2.1
Theorem 4.2.2
Theorem 4.3.2.
Theorem 4.4.1
Example 4.5.2
Example 4.6.1
Theorem 5.1.2
Example 5.1.2
Math 4032, Test # 3. Spring 2006
Name:
In problems (2)-(14) circle the number of the problem you want counted and show your
work. Recall that the function x [x] is dened by [x] = supcfw_n Z | n x.
1[15P]) True (T) or false (F). Explain your answer:
a) If
Material for Test # 3
All examples in the following sections.
Section 6.1, Problems 1,2, 4, 6,9, 14
For a proof: Example 6.1.1, Theorem 6.1.1 and Theorem 6.1.2
Section 6.2, Problems 1, 3, 4, 5, 7
For a proof: Theorem 6.2.2
Section 6.3, Problems 1, 2
Math 4032, Test # 1. Spring 2006
Name:
1[15P]) True (T) or false (F):
a) If f is dierentiable at x, then f is continuous at x. (T)
b) The series k1p converges for all p > 1. (T)
k =1
c) The series
k =1
(1)k+1 k 2
converges absolutely. (F)
2k 3 1
Answer 3
Material for Test # 2
You need all the material from Chapter 5, even if there will be no problem from Section 5.1.
You will need to know Theorem 5.1.2.
Work on the problems from Solutions to problems from Chapter 5 on my webpage. The
problems will be ta