Math 104, Fall 07
Homework#4: Compact subsets
1. Give an example for a bounded set in R which is not compact. Give an example for a
closed subset in R which is not compact.
2. Write down examples of compact and non-compact subsets in R1 , R2 and R3 :
3. S
Math 104, Spring 09
Homework#:4 Sequences
1. Show using only the denition that
(a)
(b)
1
! 0, as n ! 1:
n+1
1
! 0, as n ! 1:
n2
n
(c) 1 + ( 1) diverge as n ! 1:
2. Suppose (an ) and (bn ) are sequences of complex or real numbers. Assume that an ! a
and bn
Math 104, Spring 09
Homework#5 Sequences
1. Give an example of a metric space which is not complete, i.e., in which there exists a
Cauchy sequence which is not converge.
2. Let (X:d) be a compact metric space. Show that X is complete.
3. Let X be a comple
Math 104, Spring 09
Homework#6 Series
1. Write a proof that the series
1
P
n=2
1
n(log(n)
converge if
> 1 and diverge for
1 (You
can use the proof on page 62 in the book).
2. Find whether the following series diverge or converge:
(a)
(b)
1
P
n=1
1
P
n=1
n
Math 104, Spring 09
Homework#7: Series and power series
1. Proof that the series
1
X
1
n
diverge.
n=1
2. We dened the number e =
1
X
1
:
n
Show (you can use the book) that 1 +
n=1
n ! 1:
1
X
3. Proof the following theorem: Let
n=0
1n
n
! e as
an z n be a
Math 104, Spring 09
Homework#8: Series and continues functions
1. Let A(z ) =
1
X
an z n be a power series with radios of convergence R = 2: Assume in
n=0
addition that a1
a2
: ! 0: Show that A(z ) converge for every z with jz j = 2
except possibly for z
Math 104, Spring 09
Homework#9: Continuous functions
1. Do exercises 1.,2.,3.,4.,5.,6.,14. page 98 in the book.
2. Prove that a function f : X ! Y (X and Y are metric spaces) is continuos if an only
if for every open subset V
Y we have that the pre-image
Math 104, Spring 09
Homework#10: Connectedness, dierentiation
1. State and prove the theorem that a continuos function maps connected set to a connected set (Theorem 4.22 on page 93 of the book).
2. Let I = [a; b]
R be a closed interval with a < b: Consid
Math 104, Spring 09
Homework#11: Taylor theorem, integration
s
p
1. Use Taylor theorem to compute 3 90 with approximation which is better then 0.001.
s
2. Compute, using only the denition of integral using Riemann sums, the integral
Z1
x2 dx:
1
3. Prove t
Section 7.6: Special Techniques
Remember, oce hours are now 2:30 Tues and 10:00 Thurs.
On the notation tan1 for arctan.
0.1
Trigonometric identities
Sometimes we can use our good old trig identities in order to help us compute
integrals, by writing the
Section 7.4: Rational Functions
Oce hours today after I teach, at noon. Next week, the time will
become 2:30 pm on Tues.
The Missouri Club-you can talk about the weeks homework with loving graduate students. Meets Sunday nights, 8-11 pm, in McCosh 34.
T
Section 10.4: The integral test
Lets return to the example of the harmonic series from last time. Some of
you are still bothered that the sum
1 + 1/2 + 1/3 + 1/4 + . . .
can diverge even though the terms get smaller and smaller.
When something like
1 + 1/
Section 11.6: Complex Numbers
First of all, I want to discuss the theorem about the quotients of power
series. Observe that the power series
1 = 1 + 0x + 0x2 + . . .
and
1 + x2 = 1 + 0x + x2 + 0x3 + . . .
each converge for all x. So according to the theor
Math 104, Spring 09
Homework#3: Topology in metric spaces
1. Write down several examples of open and closed subset of R and R2 : Give examples
also for subsets which are not closed and not open.
2. Recall the denition of a limit point of a subset E of a m
Math 104, Spring 09
Homework#2: Euclidean spaces and Metric spaces
1. Problems 17, 18, 19 page 23.
2. Problem 11, page 44.
3. Draw the following sets in R2 and R3 :
(a) The open ball B (0; 1):
(b) The closed ball B (0; 1):
(c) The boundary @B (0; 1) = fx
Math 104, Fall 07
Homework#:5 Sequences
1. Show using only the denition that
(a)
(b)
1
! 0, as n ! 1:
n+1
1
! 0, as n ! 1:
n2
n
(c) 1 + ( 1) diverge as n ! 1:
2. Suppose (an ) and (bn ) are sequences of complex or real numbers. Assume that an ! a
and bn !
Math 104, Fall 07
Homework#:6 Sequences
1. Give an example of a metric space which is not complete, i.e., in which there exists a
Cauchy sequence which is not converge.
2. Let (X:d) be a compact metric space. Show that X is complete.
3. Let X be a complet
Math 104, Fall 07
Homework#:7 Series
1. Write a proof that the series
1
P
n=2
1
n(log(n)
converge if
> 1 and diverge for
1 (You
can use the proof on page 62 in the book).
2. Find whether the following series diverge or converge:
(a)
(b)
1
P
n=1
1
P
n=1
n
Math 104, Fall 07
Homework#8: Series and power series
1. Proof that the series
1
X
1
n
diverge.
n=1
2. We dened the number e =
1
X
1
:
n
Show (you can use the book) that 1 +
n=1
n ! 1:
1
X
3. Proof the following theorem: Let
n=0
1n
n
! e as
an z n be a fo
Math 104, Fall 07
Homework#9: Series and continues functions
1. Let A(z ) =
1
X
an z n be a power series with radios of convergence R = 2: Assume in
n=0
addition that a1
a2
: ! 0: Show that A(z ) converge for every z with jz j = 2
except possibly for z =
Math 104, Fall 07
Homework#10: Continuous functions
1. Do exercises 1.,2.,3.,4.,5.,6.,14. page 98 in the book.
2. Prove that a function f : X ! Y (X and Y are metric spaces) is continuos if an only
if for every open subset V
Y we have that the pre-image f
Math 104, Fall 07
Homework#11: Connectedness, dierentiation
1. State and prove the theorem that a continuos function maps connected set to a connected set (Theorem 4.22 on page 93 of the book).
2. Let I = [a; b]
R be a closed interval with a < b: Consider
Math 104, Fall 07
Homework#12: Taylor theorem, integration
s
p
1. Use Taylor theorem to compute 3 90 with approximation which is better then 0.001.
s
2. Compute, using only the denition of integral using Riemann sums, the integral
Z1
x2 dx:
1
3. Prove the
Math 104, Fall 07
Homework#1: Q, R, and C
1. Show that
p
3 2 Q.
=
2. Let (S; <) be an ordered set and E
S a set which is bounded below. Dene the
inmum of E which is denoted by = inf E: Show that there exist at most one such
:
3. Read the section "Fields",
Math 104, Fall 07
Homework#2: Euclidean spaces and Metric spaces
1. Problems 17, 18, 19 page 23.
2. Problem 11, page 44.
3. Draw the following sets in R2 and R3 :
(a) The open ball B (0; 1):
(b) The closed ball B (0; 1):
(c) The boundary @B (0; 1) = fx 2
Math 104, Fall 07
Homework#3: Topology in metric spaces
1. Write down several examples of open and closed subset of R and R2 : Give examples
also for subsets which are not closed and not open.
2. Recall the denition of a limit point of a subset E of a met
Math 104, Spring 09
Homework#1: Q, R, and C
1. Show that
p
3 2 Q.
=
2. Let (S; <) be an ordered set and E
S a set which is bounded below. Dene the
inmum of E which is denoted by = inf E: Show that there exist at most one such
:
3. Read the section "Fields
Math 104
Introduction to Analysis
Spring 2009
Instructor: Shamgar Gurevich, O ce: 867 Evans Hall, Phone: 643-7543.
Time and Location: M-W-F 3-4pm, Room 75 Evans.
Course Webpage: http:/math.berkeley.edu/~shamgar
O ce Hours:
Mon.
Wed.
1:30-2:30pm
1:30-2:30p