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    (Original post by Zacken)
    'innit. I'm putting a break on STEP till I'm done with FP3, so... I've got a few more pages and I'll be done with the whole of integration. :yep:
    Oh yeah, 'cause you are doing one STEP paper a day right? I've been doing one question a day for months now (I think) so I don't think I need to stop now. You could do one question a day as well, if you want. It will probably only take you 20-30 minutes, haha!
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    (Original post by Insight314)
    Oh yeah, 'cause you are doing one STEP paper a day right? I've been doing one question a day for months now (I think) so I don't think I need to stop now. You could do one question a day as well, if you want. It will probably only take you 20-30 minutes, haha!
    Nah, I do one STEP paper a week. Like I said, my work ethic is shite. :lol:
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    (Original post by 雷尼克)
    because i want to go there and this thread is making me cringe
    You probably won't meet your offer for Warwick anyway tbh, not sure why you're worried.
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    (Original post by Zacken)
    Took me far too long to spot the -x^2 = 1 - x^2 - 1 trick. Anyways, this requires two bouts of integration, the first one yields, with u = (1-x^2)^n and \displaystyle v = \frac{1}{\alpha}\sin \alpha x:

    \displaystyle 

\begin{equation*} \frac{\alpha I_n}{2n} = \int_{-1}^{1} x(1-x^2)^{n-1} \sin \alpha x \, \mathrm{d}x

\end{equation}

    Now, we bash it with IBP again, this time with u = x(1-x^2)^{n-1} and \displaystlye v = -\frac{1}{\alpha} \cos \alpha x which gives us:

    \displaystyle

\begin{equation*}\frac{\alpha^2}  {2n}I_n = I_{n-1} + 2(n-1)\int_{-1}^{1} \cos \alpha x (1-x^2)^{n-2} (-x^2) \, \mathrm{d}x \end{equation*}

    Clever bit:

    \displaystyle 

\begin{equation*} \frac{\alpha^2}{2n}I_n = I_{n-1} + 2(n-1)\int_{-1}^{1} \cos \alpha x (1-x^2)^{n-2} (1-x^2 - 1) \, \mathrm{d}x \end{equation*}

    So:

    \displaystyle 

\begin{equation*} \frac{\alpha^2}{2n}I_n = (2n-1)I_n - 2(n-1)I_{n-2} \iff \alpha^2 I_n = 2n(2n-1)I_{n-1} - 4n(n-1)I_{n-2} \end{equation*}

    as required. Lovely.
    Bravo! Now, show that this implies that

    \alpha^{2n+1} I_n = n! (P_n \sin \alpha + Q_n \cos \alpha)

    where  P_n, Q_n are polynomials in \alpha of degree < 2n+1 with integer coefficients.

    Hence, show that \pi is irrational. Hint: put  \alpha = \pi/2 and by assuming that \pi = b/a show that

    \displaystyle J_n = \frac{b^{2n+1}I_n}{n!}

    is an integer that tends to zero as n \rightarrow \infty
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    (Original post by Gregorius)
    Bravo! Now, show that this implies that

    \alpha^{2n+1} I_n = n! (P_n \sin \alpha + Q_n \cos \alpha)

    where  P_n, Q_n are polynomials in \alpha of degree < 2n+1 with integer coefficients.

    Hence, show that \pi is irrational. Hint: put  \alpha = \pi/2 and by assuming that \pi = b/a show that

    \displaystyle J_n = \frac{b^{2n+1}I_n}{n!}

    is an integer that tends to zero as n \rightarrow \infty



    Just kidding, looks lovely, prime procrastination material. :yep:
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    Update:

    I've put away Greg's problem/extension for a bit, I'm far too groggy to think through it properly. In any case, I think I've finished the integration chapter in FP3, I just need to work through a few more exercises now.
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    :rofl: love the title, and inspirational back story. Must take some time to read through the 9 pages.

    Good luck! (not that you'll need it...)

    :rave:
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    (Original post by SeanFM)
    :rofl: love the title, and inspirational back story. Must take some time to read through the 9 pages.

    Good luck! (not that you'll need it...)

    :rave:
    Woot! Thanks. Good to have you onboard.
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    (Original post by SeanFM)
    :rave:
    zacken zacken, he's our man if he can't do it, no one can!

    https://www.youtube.com/watch?v=emKHcHZ_A2E
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    (Original post by aymanzayedmannan)
    zacken zacken, he's our man if he can't do it, no one can!

    https://www.youtube.com/watch?v=emKHcHZ_A2E
    Brilliant. :rofl::rofl:
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      (Original post by Zacken)
      Nah, I do one STEP paper a week. Like I said, my work ethic is shite. :lol:
      That's still a question a day... and you know what they say... a question a day keeps Warwick away

      (Original post by Gregorius)
      Bravo! Now, show that this implies that

      \alpha^{2n+1} I_n = n! (P_n \sin \alpha + Q_n \cos \alpha)

      where  P_n, Q_n are polynomials in \alpha of degree < 2n+1 with integer coefficients.

      Hence, show that \pi is irrational. Hint: put  \alpha = \pi/2 and by assuming that \pi = b/a show that

      \displaystyle J_n = \frac{b^{2n+1}I_n}{n!}

      is an integer that tends to zero as n \rightarrow \infty
      Nice problem
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      (Original post by Mathemagicien)
      That's still a question a day...
      Well... not really. I do the STEP paper in one go, so nothing for 6 days then 7 questions in one day.

      and you know what they say... a question a day keeps Warwick away

      :rofl:

      Nice problem
      Have you done it?
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      Update:

      I'll do a proper update one of these days, with a nice long blog post. Anywho, think I'm going to get my FP2 June 2011 mock going now. Wish me luck!
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      (Original post by Zacken)
      Update:

      I'll do a proper update one of these days, with a nice long blog post. Anywho, think I'm going to get my FP2 June 2011 mock going now. Wish me luck!
      Good luck
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      FP2 Mock (June 2011):

      So let's do some analysis (the boring kind, not the maths kind):

      Total time: 59:20 Total raw: 75 Total UMS: 100

      Q1: inequalities. Took longer than I should because I did it in two different ways to check my answer. I'll need to improve my speed here. 9:32
      Q2: Taylor series solutions to DE's took far longer than I should have fixing a silly mistake. 6:05
      Q3: Easy first order DE, happy with my time. 3:45
      Q4: Telescoping sums, easy, took me longer than I'd like because I work slowly to avoid silly mistakes, it was an easy one as well, need to cut down on my time. 7:22
      Q5: Complex loci, my solution was far longer than need be, not pleased at all. 10:07
      Q6: Polar co-ordinates, mildly enjoyable. Did well on this one. 07:03
      Q7: Easy C3 stuff, took me long because I graphed to see if I got all my solutions. 8:07
      Q8: Easy second order DE, surprised I didn't make a silly slip, pleased with my time. 07:04

      All-around, I think I did okay. Certainly lots of space for improvement with regards to my timing issue.
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      (Original post by Zacken)
      I raped

      tl;dr for y'all.

      You're welcome
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      (Original post by Zacken)
      Total raw: 75 Total UMS: 100

      All-around, I think I did okay.
      :curious:
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      (Original post by tinkerbella~)
      :curious:

      Spoiler:
      Show
      just bants zacken <3
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      (Original post by Student403)
      Spoiler:
      Show
      just bants zacken <3
      PRSOM, this made my day :rofl:
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      (Original post by Zacken)
      FP2 Mock (June 2011):
      Total time: 59:20

      All-around, I think I did okay. Certainly lots of space for improvement with regards to my timing issue.
      Ayy lmao, top banter! :rofl:




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