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    (Original post by matt2k8)
    In your example the sequence of partial sums just oscillated between -1 and 0 so the series doesn't converge - I said it was true as it's required that (a_n) \rightarrow 0 for the series \Sigma a_n to converge, but if  lim \sup \ a_n \not= \lim \inf \ a_n then (a_n) does not converge to any limit
    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.
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    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.
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    can i request an example for the truth of 32?
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    (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
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    (Original post by yeohaikal)
    can i request an example for the truth of 32?
    a_n=\dfrac{\sqrt{2}}{2^n}
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    (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
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    When is the exam?
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    (Original post by Apex9)
    When is the exam?
    Wednesday afternoon, first week back
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    (Original post by matt2k8)
    Wednesday afternoon, first week back
    Thanks.
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    "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:
    Show
    Q1 Every non-empty subset of the natural numbers has a least element
    Q2 Every natural number greater than 1 can be expressed as a product of prime

    numbers, unique up to the order of the terms
    Q3 a|b means a divides b with no remainder, i.e. b = ma for some integer m.
    Suppose a,b,c are natural numbers such that a|b and b|c.
    Then b= ma, c =nb for some integers m,n
    Then c = mna, so as mn is also an integer, a|c.
    Q4. hcf is 1, and h = 33*52 - 7*245
    Q5. hcf is 3, and a = -4, b = 13

    Subgroups:
    Spoiler:
    Show
    Q6 n\mathbb{Z} = \{ na : a \in \mathbb{Z} \}
    n\mathbb{Z} + m\mathbb{Z} = \{ na + mb : a,b \in \mathbb{Z} \}
    i) is a subgroup, equal to 15\mathbb{Z}
    ii) is a subgroup, equal to \mathbb{Z}
    iii) is not a subgroup

    Q7 equal to 21\mathbb{Z}
    Q8 equal to 4\mathbb{Z}
    Q9 equal to 3\mathbb{Z}

    Sets, Venn Diagrams:
    Spoiler:
    Show

    Q10 (A \cup B)^C = A^C \cap B^C
    (A \cap B)^C = A^C \cup B^C
    Q11 Diagram
    Q12 (A \backslash B) \backslash  C = A\backslash (B \cup C)
    Q13 (A \backslash B ) \cap (A \backslash C) = A \backslash (B \cup C)
    Q14 truth table
    Q15 same as P \Rightarrow Q
    Q16 same as (¬\lnot P) \lor Q
    Q17 same as (¬\lnot P) \lor (¬\lnot Q)
    Q18 same as (¬\lnot P) \land(¬\lnot Q)
    Q19 same as a \land (¬\lnot b)

    Remainder Theorem:
    Spoiler:
    Show

    Q22. Let P be a polynomial. Then the remainder on division of P by x - \alpha is equal to P(\alpha)
    Q23 (x+4), (x-1)
    Q24 (x-3), (x-1)
    Q25 (x+3), (x-2)
    Q26 (2x+1), (x+4)
    Q27 (x+1)

    Pascal's Triangle:
    Spoiler:
    Show

    Q28 LHS = \frac{n!}{(k-1)!(n-k+1)!} + \frac{n!}{k!(n-k)!}
    = \frac{n!k + n!(n-k+1)}{k!(n-k+1)!}
    = \frac{(n+1)!}{k!([n+1]-k)!}
    = RHS
    Q29 Coefficient of x^k is \displaystyle \binom{n}{k}b^ka^{n-k}

    so (a+bx)^n = \displaystyle\sum_{k=0}^{n} \displaystyle \binom{n}{k}b^ka^{n-k}x^k

    Set a = 1, b = -1, x = 1 to show the result.

    Q30 same as last question, but set b = 1 instead

    Injections and Surjections:
    Spoiler:
    Show
    Q31 False. E.g. let A= \{1,2 \}, B = \{3,4 \}, C= \{3,4,5 \}, D = \{6,7,8 \}.

    Then define f:A\rightarrow B by f(1)=3, f(2)=4
    so f is a surjection, and Im(f) = B \subseteq C

    Then define g:C\rightarrow D by
    g(3) = 6, g(4) = 7, g(5) = 8
    then g is also a surjection, but g \circ f : A\rightarrow D is not a surjection as g \circ f(1) = 6 and g \circ f(2) =7 so it does not map A to all of D.
    Attached Images
  1. File Type: pdffoundations q1.pdf (107.2 KB, 238 views)
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    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!
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    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.
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    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 fixed
    Q67 - thanks, fixed it now, wasn't paying much attention as I don't know much about lim sup
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    (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.
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    Am I right in saying the past papers worth doing go back to 2006?
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    (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!

    (Original post by 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

    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
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    (Original post by placenta medicae talpae)
    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!
    It's not in any of the workbooks given to maths students, and my class tutor cut the lim sup/lim inf questions out of this activity we did doing questions from a past paper so I am 99.9% sure we don't need to know about it
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    (Original post by matt2k8)
    It's not in any of the workbooks given to maths students, and my class tutor cut the lim sup/lim inf questions out of this activity we did doing questions from a past paper so I am 99.9% sure we don't need to know about it
    Cheers; that's slightly comforting.
    Though, having had a quick glance at the stuff about it, I can't see there was anything particularly mind-bending about it or whatever to warrant removing it from the syllabus :iiam:
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    It'll probably come out in Analysis II...I wish I knew how to use LaTeX...
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    (Original post by Narev)
    I wish I knew how to use LaTeX...
    Lawl, is this a desperate attempt to hide who you are?!
 
 
 
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