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
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
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
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
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
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
Math 131A, 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