MATH3570 Main Assignment
Due: 11:59pm, Sunday 13 March 2022
Instructions:
• Which questions you need to do will depend on the last digit of your student
number.
• Your solutions to this assignment need to be properly written up using LATEX, Word
or something similar and then submitted as a PDF file via Moodle. Sample files in a
suitable format are available on Moodle.
• Submissions may be computer checked for signs of copying!
• Marks will be deducted for sloppy presentation, or imprecise logic or notation. Remem-
ber you are not doing a calculation, you are presenting an argument. Aim to get your
spelling and grammar correct (use a spell checker!) — but the clarity of the argument
is the most important point here. Write in proper sentences!
• Don’t forget to reference any sources (human/book/internet/. . . ) that you have used.
It is OK to discuss problems with your classmates as long as you record this fact. Just
make sure that you go home and write up your answers quite separately.
• You must never show them your completed solution to another student; they inevitably
copy it out and then you both get zero.
• Pay attention to the new strict late penalty rules in the course handout. (If you have
special provisions via ELS, you should remind me in the week before the assignment is
due.)
• If you are really stuck on something, feel free to email me.
• If a question needs clarification, or you think that it is wrong, please let me know!
Which questions do I do?
Every student needs to hand in the solutions to 4 questions. The particular questions you
need to hand in are given in the table below. (You should of course try the other questions
if you get the time to do so!) Anyone getting Question 1 wrong will be severely penalized!
Last digit Questions to hand in
0 1,2,4,6
1 1,2,4,7
2 1,2,5,6
3 1,2,5,7
4 1,3,4,6
5 1,3,4,7
6 1,3,5,6
7 1,3,5,7
8 1,2,4,6
9 1,3,5,7
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Questions:
1. Complete the following sentence:
The last digit of my student number is . . . so I need to do questions . . .
For the next two questions, the following fact might be helpful: if {αn} is a decreasing
sequence of positive numbers then
∣∣∣∑`n=k(−1)nαn∣∣∣ < αk. You don’t need to prove this
(but you should understand geometrically why it is true!).
2. Consider the statement
∀N > 100 ∀x ∈ (0, N/2) ∃y ∈ (x,N),
∫ y
x
cosu
u
du = 0.
(a) Write the negation of the above statement.
(b) Which is true, the statement or its negation? Prove your answer.
3. Consider the statement
∃s > 0 ∀a ≥ s ∀b > a,
∣∣∣∫ b
a
sin t
t
dt
∣∣∣ ≤ 1.
(a) Write the negation of the above statement.
(b) Which is true, the statement or its negation? Prove your answer.
4. Consider the sequence a0 =
1
2
, an+1 =
3an + 2
2an + 4
, n ≥ 0.
(a) Prove that {an} converges.
(b) What are s = sup
n≥0
an and t = inf
n≥0
an?
(c) Can you get different limiting behaviour by starting at a different value a0?
(I want more than ‘Yes’ or ‘No’ here — but not necessarily full proofs!)
5. Consider the sequence b0 = 1, bn+1 = ln(bn + 2), n ≥ 0.
(a) Prove that {bn} converges.
(b) What are s = sup
n≥0
bn and t = inf
n≥0
bn?
(c) Suppose that we change the recurrence to b0 = 0.5, bn+1 = 3.5(bn − b2n) (n ≥ 0).
What is the limiting behaviour of this sequence? (No proofs required.)
Please see over . . .
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6. We are going to apply the bisection procedure from Section 3.8. Start with the interval
I0 = [0, 1]. The rule for choosing subintervals In = [an, bn] is:
• If n is a power of 2, then let In be the right half of In−1.
(Note: we’ll consider 1 = 20 as a power of 2.)
• If n is not a power of 2, then let In be the left half of In−1.
(a) Find I1, I2 and I3.
(b) What is a12?
(c) Let c = lim
n→∞
an. What are the first 6 decimal digits of c. (Explain how you know
that your answer is correct!)
7. Suppose that f, g : [−1, 1]→ R are continuous functions. Setting
d(f, g) = sup
t∈[−1,1]
|f(t)− g(t))|
gives a useful measure of the ‘distance’ between the two function. (You don’t need to
prove this, but, for example, it satisfies the ‘triangle inequality’: d(f, g) ≤ d(f, h) +
d(h, g) for any continuous function h.)
For x ∈ [−1, 1] define
F (x) = sin(x),
G(x) = cos(x)
and for n = 1, 2, 3, . . . , let hn(x) = tan
−1(nx), x ∈ [−1, 1].
(a) Calculate d(F,G).
(b) Prove that for all x ∈ [−1, 1], the sequence {hn(x)}∞n=1 converges in R.
(c) Prove that {hn}∞n=1 is not a Cauchy sequence with respect to the distance d (see
Section 3.11 of the notes).
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