131A Homework 4- Due Friday, October 21.
Chapter 1 # 11.1,11.2,11.4,11.6 And the following exercises:
1. Suppose (an )
n=1 is a sequence and that an cfw_0, 1, . . . , 9 for each n N.
P
k
Prove that the series
k=1 10 ak is Cauchy.
What is the point of thi
Math 131A, Fall 2016
Practice Midterm
Version 1
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First name
Student ID
Use the provided space for your solutions (you may use the additional page at the end). Show your work. Write only one solution
per problem - problems with multiple solutions
Math 131A: Preparation Questions for Quiz 1
1. (a) Prove the following sentence is true.
2
2
a R \ cfw_0, b R, c R, b 4ac 0 = x R : ax + bx + c = 0 .
(b) Prove the following sentence is true. (Id save this until after 2, 3, 4.)
a R \ cfw_0, b R, c R,
x R
Math 131A: Homework 5
Please turn this homework in, to me, at the start of lecture on March 10th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz for this
homework is on March 16th.
1. (a) Suppose f :
Math 131A: Homework 4
Please turn this homework in, to me, at the start of lecture on February 24th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz for this
homework is on March 2nd.
1. Let f : R R b
Math 131A: Homework 2
Please turn this homework in, to me, at the start of lecture on January 27th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz for this
homework is on February 2nd.
1. Let S and T
Math 131A: Homework 3
Please turn this homework in, to me, at the start of lecture on February 10th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz for this
homework is on February 16th.
1. Let (sn )
131A Homework 5- Due Friday, November 3.
Chapter 3 # 17.1,17.2,17.3,17.417.7,17.8,17.11,17.15 And the following exercises:
1. It is true that limx3 (x3 + 1) = 28.
(a) Verify the sequence definition using the algebra of limits .
(b) Verify the - definition
Math 115A, Fall 2016
Practice Midterm
Version 1
Last name
First name
Student ID
Use the provided space for your solutions (you may use the additional page at the end). Show your work. Write only one solution
per problem - problems with multiple solutions
Analysis, Math 115A, Fall 2016 - Course Info
Instructor: Martin Tassy, mtassy@math.ucla.edu,
Instructor Office Hours: WTh 9-10am in 6324 Science Building
Lectures: Monday, Wednesday and Thursday, 11am-11:50am in MS 5117
Teaching Assistants: Hoafei Fan, fh
HOMEWORK 3: SOLUTIONS/HINTS
8.4 Let > 0, then there is an N such that |sn | < M for n N . So we get that |sn tn | M |sn | < for all n N and hence lim(sn tn ) = 0. 8.5 Let > 0, then we have N N such that |an - s| < as well as |bn - s| < for n N. So for n N
HOMEWORK 4: SOLUTIONS/HINTS 10.1 (d) Let sn := sin( n ), then |sn | 1, hence the sequence is bounded. 7 However, it is neither increasing nor decreasing. (f) Let an := 3n . Then an > 0 for all n, so the sequence is bounded n below. Moreover, n+1 an+1 1 1
131A Homework 3- Due Friday, October 14.
Chapter 1 # 9.1,9.4,9.9,9.12,9.15,
10.1,10.6,10.10,10.12
And the following exercise:
Suppose (sn )
n=1 is a sequence such that for each k N with k 2, the subsequence (skn )n=1
converges. Either
prove that (sn ) co
131A Homework 1- Due Friday, September 30.
Chapter 1 # 1.1,1.3,1.6,1.10,1.11,4.1,4.7.4.10
And the following exercice:
For each n N, let Pn denote the assertion n2 + 5n + 1 is an odd integer.
1. Prove Pn+1 is true whenever Pn is true.
2. Show that Pn is al
Math 131A: Homework 1
Please turn this homework in, to me, at the start of lecture on January 13th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz for this
homework is on January 19th.
1. (a) Without