First year Analysis/Foundations January Exam 2011

i think u're missing out a power of n on the -1 for the solution in qns 6. btw, a_n=ln(n)/n is another solution since a_2=a_4 and a_3 is greater than both.

Reply 22

can i request an example for the truth of 32?

Reply 23

OP

Original post by yeohaikal

can i request an example for the truth of 32?

a_n = 2 + \frac{\sqrt{2}}{n}

is always irrational but tends to 2. Fixed question 6 as well btw, thanks :smile:

Reply 24

miml

17

Original post by yeohaikal

can i request an example for the truth of 32?

a_n=\dfrac{\sqrt{2}}{2^n}

Reply 25

OP

Original post by yeohaikal

Q5 is true.. by giving a strict inequality, it just gives a stronger (sufficient) definition of not null, though definitely not necessary. is that right?

Yeah that condition is definitely sufficient to show it's not null but I'm not sure about if for it to be considered a definition it also has to be a neccasary condition

Original post by yeohaikal

am i have a conceptual error if i say that 'a conditionally convergent sequence is convergent'? having said that, i can see your point. just would like to clarify.

The phrase conditionally convergent refers to series where

\Sigma |a_n|

diverges but

\Sigma a_n

converges, not anything to do with sequences

(edited 13 years ago)

When is the exam?

OP

Original post by Apex9

When is the exam?

Wednesday afternoon, first week back

Reply 28

Apex9

8

Original post by matt2k8

Wednesday afternoon, first week back

Thanks.

Reply 29

OP

foundations q1.pdf109.7KB

"Question 1" Foundations Questions from MORSE Booklet - left out the induction section as it's easy to tell if it's right or wrong.
Questions up to 31 are in spoilers below - 32 are in PDF file attached
Basics:

Spoiler

Subgroups:

Spoiler

Sets, Venn Diagrams:

Spoiler

Remainder Theorem:

Spoiler

Pascal's Triangle:

Spoiler

Injections and Surjections:

Spoiler

(edited 13 years ago)

Reply 30

Ok.. Here's a list of a few more stuff I want to clarify:

Q33. FALSE. Can I have a counterexample? Because I thought the restriction of a_n being positive decreasing and null makes it true?

Q45. FALSE. I think False is right but your counterexample is not a correct counterexample because a_n tends to 1? I think we are looking for a_n that does not converge. My counterexample is a_n=(n-2)^2. Since there exists an N=2 s.t. abs(a_2-0)=0<epsilon for all epsilon>0, but a_n does not tend to 0.

Q67. I know that it's out of syllabus, but just for discussion sake, doesn't the example a_n=10+(1/n) show that it's false?

thats abt it. anyways, thx for putting up e solns!

Reply 31

W.R.T. Foundations questions, here's some things I want to point out..

1.11. You're missing the other half of De Morgan's Laws. They come as a pair. (A union B)^c=A^c intersect B^c. help me do the latex-ing please. :P

1.22. Definition of Remainder theorem is missing out the fact that we are only talking about the reals.. Or does it not matter? Because I'm sure there are some values of P(alpha) that do not exist in say the naturals or integers or rationals.

Reply 32

OP

Have let one of my friends borrow my answers until tommorow so no more will be typed tonight

Original post by yeohaikal

Ok.. Here's a list of a few more stuff I want to clarify:

Q33. FALSE. Can I have a counterexample? Because I thought the restriction of a_n being positive decreasing and null makes it true?

Q45. FALSE. I think False is right but your counterexample is not a correct counterexample because a_n tends to 1? I think we are looking for a_n that does not converge. My counterexample is a_n=(n-2)^2. Since there exists an N=2 s.t. abs(a_2-0)=0<epsilon for all epsilon>0, but a_n does not tend to 0.

Q67. I know that it's out of syllabus, but just for discussion sake, doesn't the example a_n=10+(1/n) show that it's false?

thats abt it. anyways, thx for putting up e solns!

Q33 -

a_n = 1/n

Q45 - thanks, don't know what I was thinking with that one haha :smile:

fixed
Q67 - thanks, fixed it now, wasn't paying much attention as I don't know much about lim sup

(edited 13 years ago)

Reply 33

OP

Original post by yeohaikal

W.R.T. Foundations questions, here's some things I want to point out..

1.11. You're missing the other half of De Morgan's Laws. They come as a pair. (A union B)^c=A^c intersect B^c. help me do the latex-ing please. :P

1.22. Definition of Remainder theorem is missing out the fact that we are only talking about the reals.. Or does it not matter? Because I'm sure there are some values of P(alpha) that do not exist in say the naturals or integers or rationals.

1.11 Thanks, fixed now, was having troubles with LaTeX so forgot it haha

1.22 it doesn't matter, in the lecture notes it says it's true as long as the coefficients belong to any field.

Reply 34

Apex9

8

Am I right in saying the past papers worth doing go back to 2006?

Reply 35

placenta medicae talpae

Original post by yeohaikal

Q67. I know that it's out of syllabus

So we don't need to know about lim-sup and lim-inf?
It worried me quite a lot, when reading about them in the maths society revision guide, because I was sure we hadn't even had those words so much as mentioned in lectures! :biggrin:

yeohaikal

(A union B)^c=A^c intersect B^c

For the LaTeXing, all you need is to put [&latex&] and [&\latex&] at the beginning and end, removing the ampersand 'and' signs.
And then everything you've written is cool, apart from change union to \cup and intersect to \cap :h:

In fact, if you hover over (or click on) any LaTeX output on here, it displays the input, which is quite handy.
(A \cup B)^c=A^c \cap B^c

(edited 13 years ago)

Reply 36

OP