Solutions to Final Exam. Math 2J. Summer Session II, 2013.
1. Determine whether the following sequences converge or diverge and nd the limits of the convergent ones.
Show your work.
5n2 3n
2n2 + 1
n! + 1
2n2 + 3
1
c) an = 2 + 3 cos n
d) an = log(sin )
n
N
Math 2J Final
ID:
(Dec. 5th, 2011)
Name:
1. (12) True or false. (No proof or calculation is required.)
(a) If lim an = 0, then lim an  = 0.
n
(b) If 0 an bn and
n
bn converges, then
an converges.
(c) If cfw_an is increasing and an < 6 for all n, then c
Problem 1 (10 points). Evaluate the following limits. JUSTIFY your answers. If a limit
does not exist, say so.
2+3n2
(1) limn 3+2n2
(2)
(3)
(4)
(5)
2
n)
limn (lnn
limn sin(n)+n
2n
limn (1)n en
limn sin n+3
2n
Solution.
(1) This converges to 3/2. You can d
Problem 1 (6 points). Find the general solution of the following system of equations. If it is
inconsistent, say so.
x1 + x2 + x3 x4 = 3
x1 x2 x3 x4 = 1
2x1 2x4 = 2
Solution. Use Gaussian elimination on the augmented matrix of the system:
1 1
1 1 3
1 1
Final Review
I.
II.
III.
Sequence
Find the limit of a sequence.
Squeeze theorem (what is the theorem, how to apply it).
Monotonic sequence theorem (what is the theorem, how to apply it).
Series
Definition of converge
Final
Math 2J, Fall 2011
December 7, 2011
Instructor: Christoph Weiss
Full name:
Student #:
You have 75 minutes to complete the test. You are not allowed to use any aid but a pen(cil).
If you need additional sheets of paper, raise your hand and we will gi
Math 2J, Winter 2012
Practice problems for the midterm
1. Determine whether the statement is true or false.
(a) Overdetermined systems of equations always have no solution.
(b) Row equivalent matrices have equal determinants
2. (a) Give an example of an u
Math 2J Quiz #5  8am session
Name:
Student ID:
Problem 1. For each of the following series, decide whether the series converges.
Give reason.
(1)
n=1
10
101010
n2 4n
10
10
! n2 + 101010 ! n+ 101010 !
;
1010
Solution 1. For convenience, let us set K = 101
Math 2J Quiz #4  8am session
Name:
Student ID:
Problem 1. For each of the following sequences, decide whether the sequence converges. Give reason.
(1) an =
n
n! ;
Solution 1. Observe that
n
n
1
1
an =
=
=
=
.
n!
n(n 1) (2)(1)
(n 1)(n 2) (2)(1)
(n 1)!
On
Math 2J Final
ID:
(Dec. 5th, 2011)
Name:
1. (12) True or false. (No proof or calculation is required.)
(a) If lim an = 0, then lim an  = 0.
n
n
true
(b) If 0 an bn and
bn converges, then
an converges.
true
(c) If cfw_an is increasing and an < 6 for all
Proof of 5.
Let, D = (A+B)C, E=AC+BC ,
n
dij = ! (aik + bik )ckj ,
k=1
n
n
n
k=1
k=1
k=1
eij = ! aik ckj + ! bik ckj = ! (aik + bik )ckj = dij .
Hence D=E, which proofs relation 5.
Proof of 7.
Let, D = ! (AB), E=(! A)B, F=A
Math 2J: Quiz 6 (Feb 19th2013)
Name and id:
1. (6pt.) Let an =
5n2 + 3n + 1
. Find the limit of this sequence (show work).
1 + 3n + 60n2
2. (2pt.) Recall that given p > 0 and tn = 11p +
p > 1.
Let q > 0, and let sn (n > 0) be dened by:
1
2p
1
+ . . . np
Math 2J: Quiz 2 (Jan.152012)
Name and id:
1. (5pts.) Find all the solutions to the following linear system:
2
1
3 0
1
2 0
S: 1
3 1 4 0
Questions 26 are multiple choice: no justication is needed.
2. (2pt.) Let R be the following system:
1
2
3 0
R : 1 2 3
Math 2J: Quiz 3 (Jan.222012)
Name and id:
1. (4pts.) Find the inverse (if possible) of the following matrix A, and verify your answer by
checking that AA1 = I:
1
0
0
1
0
A= 2
3 1 1
Questions 27 are multiple choice: no justication is needed.
2. (2pt.) Le
Math 4: Practice Problems
1. Complete the following information:
Linear ISLM Model (2equations x 2variables Linear system)
Pages on Book:
Meaning of variables: R=
, Y=
,
Parameters: , , , .
Model: (IS): R = Y , (LM): R = + Y .
Matrix form:
R
=
Math 2J: Quiz 7 (Feb 26th2013)
Name and id:
Answers 1( ), 2(
), 3(
), 4(
), 5(
), 6( ), 7( ).
All the questions are multiple choice: no justication is needed. Please write your answer letters
above, by lling the spaces.
an
7 5n
46
1. (2pt.) Let an = 42n+
1. a) Solve, using Gaussian elimination, the system
Ax : b,
Where
1 1 1 1
A : 2 3 ~4 , b : 9
1 1 1 W1
5) Find A1 (Where A is that from part ((1)). Solve the system
0
Ax: O
1
using the matrix 141.
Show all your work.
[6175]
I
 4 4 4 1
a} 4 4 1 :1 :1
2
Math 2J Exam II: Practice Version
September 8, 2012
Please try the exam by yourself rst before looking things up or asking anybody for help. This will help you identify what you need to study. You must
show your work unless the problem states otherwise. N
Math 115 HW #3 Solutions
From 12.2
20. Determine whether the geometric series
en
n=1
3n1
is convergent or divergent. If it is convergent, nd its sum.
Answer: I can rewrite the terms as
en
3n1
Therefore, the series
en
n=1
3n1
=e
en1
e
=e
n1
3
3
=e
n=1
e
3
Setion 2.3  Solution of the homework
1. Exerise 3:
The sequene (a ) is onvergent sine:
n
lim a = lim
n
4 + 2=n
4n + 2
= lim
4n + 1
4 + 1=n
4
= 1:
4
P
We then have lim a = 1. This implies that the series a is not onvergent sine the
limit of a is not zero.
110
4.3
CHAPTER 4. SEQUENCES AND LIMITS OF SEQUENCES
Limit of a Sequence: Theorems
These theorems fall in two categories. The rst category deals with ways to
combine sequences. Like numbers, sequences can be added, multiplied, divided,
. Theorems from thi
Math 2J: Linear Algebra
Andres Forero Cuervo, aforeroc at uci.edu
Practice Sample Test B: Eigenvalues and Taylor polynomials
1. Using geometric arguments, determine the eigenvalues and eigenvectors for the following 22
matrices (justify briefly):
(a) 0, t
Section 2.1  Solution of the homework
1. Exercise 1:
Let us rewrite (an ) as:
an =
2n 3
2/n 3/n2
=
.
n2 + 1
1 + 1/n2
We know that 1/n and 1/n2 converge to zero, since:
1
  < ,
n
for any > 0, if n >
Also:
1
(we can decide to call this number N = 1 ).

Setion 4.2  Solution of the homework
1. Exerise 1:
(a) The series is divergent sine the limit of the term of the series does not exist:
lim( 1)n n n does not exist (pik the subsequenes with n even and n odd and
notie that the limit are dierent).
(b)
+3
1