THE UNIVERSITY OF MANITOBA
March 11, 2011
DEPARTMENT & COURSE NO: Mathematics 2730
EXAMINATION: Sequences and Series TIME: 1 HOUR
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TERM TEST 2
PAGE NO: 1 o
UNIVERSITY OF MANITOBA
DATE: March 12, 2014
MIDTERM 2
PAGE: 1 of 5
TIME: 75 minutes
EXAMINER: Harland
DEPARTMENT & COURSE NO: MATH 2730
EXAMINATION: Sequences and Series
1. For the series:
n=1
(1)n+1
n5
[4]
(a) Show the series is convergent.
Since cfw_bn
MATH 2730 Assignment 1
Solutions
1
=0.
n ln(n + 1)
Solution. We need to show that for every > 0 , there is a number M, such that for every
1
n > M , we have
0 < . We consider the last inequality first, simplifying as
ln(n + 1)
much as possible:
1
1
1
0 <
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
MATH 2730 Assignment 3
Due March 10, 2008, (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)
(1)
(c)
n +1 n
10
n
MATH 2730 Sequences and Series
Midterm Exam
5:30-6:30, February 27, 2008
Brief Solutions
1.
(a) Complete the following definition: A sequence cfw_an =1 converges to a number L
n
(written lim an = L ) if .
n
(b) Use only the definition of the limit of a se
1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
Justify your answer.
(a)
n=1
(b)
n=1
(c)
n=3
(d)
n
(1)n 5+n
sin(4n)
2n
[8]
[8]
(1)n ln n
n
[8]
an where a1 = 1, an+1 =
n=1
2+sin n
an
n
(n 1)
[8]
2. Find the
FINAL EXAM OF DEC. 2009
(1) Determine whether the series is absolutely convergent, conditionally convergent, or
divergent. Justify your answer.
(a)
(b)
(c)
n=1
n=2
n=1
1
(n + 1) 2 ( 1 )n+1
2
(1)n
n
1 n
[8]
[8]
1
(1)n e n
n
[8]
(2) Evaluate the indenite in
THE UNIVERSITY OF MANITOBA
February 7, 2011
DEPARTMENT & COURSE NO: Mathematics 2730
EXAMINATION: Sequences and Series TIME: 1 HOUR
TERM TEST 1
PAGE NO: 1 of 11
EXAMINER: Kalajdzievski
Values
[8] 1. (a) [2] State precisely the definition of a convergent s
UNIVERSITY OF MANITOBA
DATE: February 6, 2014
MIDTERM 1
PAGE: 1 of 7
TIME: 75 minutes
EXAMINER: Harland
DEPARTMENT & COURSE NO: MATH 2730
EXAMINATION: Sequences and Series
[6] 1. Find the limit of the sequence
4
n
n sin
+
n2 + 3
3n 4n2
.
n1
This can be sp
Winter 2008 Final (Warning: This test was only 2 hours long, not 3. Hence its not
indicative of the length)
1. Determine if the sequence cfw_an converges of diverges. In the case it converges, nd the
limit.
1 + 3n
5n+1
(b) an = n + 1 n
n!
(c) an = n
4
(a
Math 2730 Assignment 4
Solutions
1. Find the power series with sum equal to g(x) =
x2
and find the interval of
(1+ 2x) 2
convergence of the power series.
Solution. First we focus on the function h(x) =
1
. Observe first that
(1+ 2x) 2
1
1 1
= (2x) n and
1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
Justify your answer.
(a)
n=1
(b)
n=1
(c)
n=1
2
+1
( 2n 2 )n
3n2
2
1+n+n
1+n3 +n6
1
cos(n) sin( n )
[4]
[4]
[4]
2. Find the sum of the series
n=1
3
n(n+3)
if