2. 136.271: Test #2
20 minutes
Name:_ Student Number: _
[8] 1. Find an equation of the line passing through P(1,2,3) and parallel to the line
given in parametric form by
.
The vector (2,-1,0) is parallel to the given line (we read it from the equation: ju
136.271 Assignment 1
Solutions
1. Use only the definition of the limit of a sequence to show that lim
n
3
1 n
+2= .
2
2n 2
Solution. The question is easier than intended. The point is that
1 n
1
3
1 n
+2=
+2=
+ 2 = , so that the problem reduces to showing
136.271 Midterm Exam 2004, February 27, 4:30-5:30
Solutions
1. [2] (a) State the definition of the limit of a sequence. That is, what does it mean to say that the limit of a sequence cfw_an is the number L?
1- n [6] (b) Use only the definition of the lim
136.271 Assignment 4
Solutions
1. Find the Maclaurin series representation for the following functions and identify the
interval of convergence of the series.
2
(a)
e 2x 1
x2
(b)
sin x cos x (Hint: start with sin2x )
(c)
tan1 (3x)
2
xn
(2x 2 ) n 2 n x 2n
136.271 Assignment 3
Due March 13, 2004, (Solutions)
1. Which of the following series converges absolutely, which converges conditionally
and which diverges? Justify your answers.
(1) n +1 (0.1) n
n
n=1
n +1
(2)
n + 5n
n= 2
(a)
(b)
(c)
(1)
n +1 n
10
n=
136.271
Midterm Exam 1
SOLUTIONS
February 27 2003
(60 minutes; justify your answers unless otherwise stated; no calculators)
Note: the marks for the questions add up to 65, for 32.5% of your mark: the
extra 2.5% are bonus.
1. [13]
(a) Finish off the follo
136.271
Assignment 3: Solutions
1. [6 marks] Find the sums of the following series.
(a)
n(n - 1) x
n
, x < 1.
n =2
(b)
n2 - n
2 2n .
n=
Solution.
n =2
(a)
n =2
n =2
n =0
n(n - 1) x n = x 2 n(n - 1) x n - 2 = x 2 ( x n ) = x 2 ( x n ) , where the last eq
136.271
Assignment 1 Solutions
3n - 1 3
= by using the definition of a convergent sequence
n 4 n + 2
4
and no other properties of sequences.
1. [6 marks] Show that lim
Solution. Take an arbitrary e >0. We want to show that there is an N such that if n>N
t
136.271
Assignment 4: Section 9.8 and Uniform Convergence
(Due April 7 in class)
1.
(a) Use the binomial series to expand x (1 - x ) -2 . Simplify your answer.
n
(b) Use part (a) to find the sum of the series n . (No marks if other methods
n =1 2
are used
136.271 Assignment 2
Solutions
Note before you start: there are many ways to solve the problems below, and I do
not claim the solutions below are the shortest. They are just the first to come.
1. Use the integral test, the (simple) comparison test, the li
1. 136.271: Test #1
20 minutes
Name:_ Student Number: _
1. Suppose
and
. Find
(a)
(b)
(c)
2. Suppose
(a) Find the unit vector in the direction of .
. So the unit vector in the direction of
is
.
(b) Find one vector which is perpendicular to .
We just need
1. 136.271: Test #3
20 minutes
Name:_ Student Number: _
[8] 1. Solve the following system using Gauss-Jordan elimination (no marks if other
methods are used).
The augmented matrix is
echelon form:
. We use row reduction to find its row reduced
The system
Mini quiz #2
(Sept.25.2000)
1.
Find the sum of
answer.)
.(Do not simplify your
Solution:
2.
It is known that
converges. Find
.
Solution:
since
converges.
3. True or false ? (Do not justify your answer).
(a) If
converges then
converges. True
(b) If
converg
Miniquiz#1
(Sept.14.2001)
Considerthefollowingsequences.
A.
B.
1,1,1,1,1,1,.
C.
D.
E.
Whichofthesesequencesis
a.
decreasing?
(A)and(D)
b.
bounded?
(A),(B)and(D)
c.
monotonic?
(A),(C)and(D)
d.
divergentto
(C)and(E)
e.
convergent?
(A)and(D)
?
Mini Exam (bonus)
1
Suppose a1 = 2 and an +1 = an for n=1,2,. Is the sequence cfw_an convergent ?
Justify your answer in not more than two sentences.
2n
2 The sum of the series n +1 is
n =0 3
2
B.
C. 1
A.
3
3
The Ratio Test applied to
D.
3
2
E. None of
136.271
Midterm Exam 1
October 17, 2001
(50 minutes; justify your answers unless otherwise stated; no calculators)
1. (a) State the definition of a convergent sequence. (What does it mean to say
that
?)
For every
there exists a number N such that if
then
136.271
Midterm Exam 2 Solutions
November 21, 2001
(50 minutes; justify your answers unless otherwise stated; no calculators)
1.
Find the interval of convergence of the series
(3x - 2) n
n 3n
n =1
.
(3x - 2) n +1
Using the ration test: lim
n
n
1
( n + 1)
136.271
Midterm Exam 2
November 17 2000
(50 minutes; justify your answers unless otherwise stated; no calculators)
1.[5] Find the radius of convergence and the interval of convergence of the
series
Use the ration test:
The series converges if
<1, which me
136.271
Assignment 2 (Sections 9.3, 9.4 and 9.5)
SOLUTIONS
1. [9 marks]
(a) Use the Integral Test to test if the series
ne
-n 2
converges. Do not forget to
n =1
check that the Integral Test is applicable before you apply it.
1
(b) Use the Comparison Test