MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 2
Due in: Monday 25th September, 11:30 AM
On this sheet, all sequences are sequences of real numbers. Please hand in
solutions to questions 1-3. Question 4 is for interest only feel free
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 3
Due in: Monday 2nd October, 11:30 AM
Please hand in solutions to questions 1-3. Question 4 is for interest only
feel free to collaborate on it or ask me about it.
Compulsory questions
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 8
Due in: Monday 20th November, 11:30 AM
Please hand in solutions to questions 1-3.
Compulsory questions
1 Recall that the Fourier series for f (x) = x when
x < , and f
2-periodic is 2 n
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 9
Due in: Monday 27th November, 11:30 AM
Please hand in solutions to questions 1-3. Where appropriate, you may quote
the solutions to the heat equation and the wave equation given in lect
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 1
Due in: Monday 18th September, 11:30 AM
On this sheet, all sequences are sequences of real numbers. Please hand in
solutions to questions 1-3. Question 4 is for interest only feel free
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 4
Due in: Wednesday 11th October, 11:30 AM
Please hand in solutions to questions 1-3. Question 4 is for interest only
feel free to collaborate on it or ask me about it.
Compulsory questi
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 6
Due in: Monday 6th November, 11:30 AM
Please hand in solutions to questions 1-4. Question 5 is for interest only
feel free to collaborate on it or ask me about it.
Compulsory questions
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 5
Due in: Monday 16th October, 11:30 AM
Please hand in solutions to questions 1-4. Question 5 is for interest only
feel free to collaborate on it or ask me about it.
Compulsory questions
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 7
Due in: Wednesday 15th November, 11:30 AM
Please hand in solutions to questions 1-3.
Compulsory questions
1 Find the Fourrier coecients for the following functions. You may use
either f
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 1
Model Solutions
Compulsory questions
1 Prove from the denition of convergence that the sequence 1, 2, 3, . . . does
not converge to any real number x.
We need to show that for any x,
(
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 6
Model Solutions
n
2j
n
j
2
j =0 (1)
1 Show that cos n =
cosn2j sin2j and that sin n =
n
cosn2j 1 sin2j +1 . [Hint: use ei = cos +
2j + 1
i sin and the binomial formula.]
n
j
2
j =0 (1)
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 5
Model Solutions
Compulsory questions
1 (a) Find the radius of convergence of
The ratio between consecutive terms is
is 2 by the ratio test.
n
(1)n x2n
n=0
4n+1 .
x2
4,
so the radius of
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 4
Model solutions
Compulsory questions
1 Which of the following series of functions converge uniformly on the interval (0,1)? If they do not converge uniformly, is the limit continuous?
(
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 2
Model Solutions
1 Which of these series converge? In each case, determine whether the convergence is absolute. Justify your answers.
(a)
n n2 +3
n=0 (1) n3 7n+4
(n2 + 3)(n + 1)3 7(n + 1
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 3
Model solutions
n
2
1 Dene the sequence an recursively by a0 = 1, and an = i=1 (iani for
+2)!
1
n
1. Given that
n=0 an converges, show that
n=0 an = 2(3e) .
1
[Hint: Take the Cauchy pro
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 7
Model Soultions
1 Find the Fourier coecients for the following functions. You may use
either f (x) = n= cn einx or f (x) = 1 a0 + n=1 an cos nx + bn sin nx.
2
1 if (2n 1) < x 2n
1
if 2n
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 8
Model Solutions
1 Recall that the Fourier series for f (x) = x when
x < , and f
2 -periodic is 2 n=1 (1)n+1 sinnnx . By integrating this 4 times, nd the
23
x5
x
4
Fourier series for g
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Mock Midterm Examination
This is intended to be of a similar style to the midterm exam. Since it has
about the same number of questions as the midterm, it was not possible to
include questions covering
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Mock Midterm Examination
Model Solutions
1 Show that if the series
so does n=0 a2 .
n
n=0
an converges, where an
0 for all n, then
Since n=0 an converges, we must have an 0 as n . Therefore, we
can choo
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Midterm Examination
Wednesday 25th October: 18:0019:30
Model Answers
Answer all questions.
1 Which of the following series converge? For series which converge, is
the convergence absolute? Justify your
MATH 3090, Advanced Calculus I
Fall 2006
Toby Kenney
Homework Sheet 9
Model Solutions
Compulsory questions
1 An rod of length and 0 thickness is heated to a uniform 100 C at time
0. The ends of the rod are then immersed in ice, to x their temperature
at 0
MATH 3090, Advanced Calculus I
Fall 2006
Final Examination
Model Solutions
1 Which of the following series of functions converge uniformly on the interval (0,1)?
(a)
1
n=1 (x+n)2
1
1
As x > 0, (x+n)2 < n2 , so this series converges uniformly by the Weier1