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Binomial expansion is valid when.... (C4)

Discussion in 'Mathematics' started by rustybug, Feb 3, 2011.

  1. I am just having a bit of a brain strain, and wondered if anyone would mind helping me... In C4, in binomial expansions, if you are expanding eg (1-x/2)^n, at the end you say the expansion is only valid when |x/2| < 1 and if you are expanding (1+x/2)^n, again you say at the end it is only valid when |x/2| < 1 . I used to find this obvious but haven't thought about for a while and can't figure it out now!
    What obvious thing am I missing?
     
  2. I am just having a bit of a brain strain, and wondered if anyone would mind helping me... In C4, in binomial expansions, if you are expanding eg (1-x/2)^n, at the end you say the expansion is only valid when |x/2| < 1 and if you are expanding (1+x/2)^n, again you say at the end it is only valid when |x/2| < 1 . I used to find this obvious but haven't thought about for a while and can't figure it out now!
    What obvious thing am I missing?
     
  3. The terms of the infinite series involve (x/2)^N
    If x=1.999999... this part will fade to 0 as N tends to infinity, hence a finite sum to the series . But if x= 2.00000....1 the power term tends to infinity as N tends to infinity hence the series is unstable and not finite.
     
  4. Hi
    Bolded is the bit that needs changing
    x/2 leads to |2|<x
    3x leads to |1/3|<x
     
  5. David Getling

    David Getling Senior commenter

    Bad eg. This produces a finite series and so the expansion is valid for all x.

    To get back to finding this obvious again, read up on the ratio test. Also, rather than prescribing an interval that's open at both ends, you should check each end point.
     
  6. bombaysapphire

    bombaysapphire Star commenter

    Only if n is a positive integer. In C4 n can be any real value.
     
  7. 'To get back to finding this obvious again, read up on the ratio test. Also, rather than prescribing an interval that's open at both ends, you should check each end point.'
    Yes, in general this is good advice - eg necessary for IB option on analysis - but for C4 I think the interval is always open for the binomial series.
    Tell me if I am wrong; in fact I'm starting to doubt my own words!
     
  8. Yes that really was a brainstrain moment, I didn't even pose the question properly! I was thinking of n<0 and didn't say it, and of course I meant "condition for convergence", though actually the text book does use the phrase "expansion valid for".
    Thanks for the answers and for the link - interesting reading!
     
  9. It is odd to think that loads of us are all teaching the same thing at the same time

    Anyone else hit the stumbling block of the Y12 teacher not really teaching about n!/(n-r)!r! and just getting them to use nCr on the calculator so they do not have the starting knowledge [​IMG]
     
  10. As a year 12 teacher who has just finished planning C2 binomial expansion lessons for next week, time is so tight that I have a maximum of 15 mins to teach and have them practice n!/(n-r)!r! No guarantees that they will remember this come yr 13!
     
  11. A lack of recall is different from them not having been taught

    If they have only been told to use the nCr button with no explanation then they are at a disadvantage when I have the limited time to teach the next step at A2
     
  12. David Getling

    David Getling Senior commenter

    For certain schools and maths departments, in New Zealand, I found this was a big problem. Lots of students were being taught to get answers by simply pressing the right buttons on their graphics calculators. I've absolutely no doubt that for some students the only difference between a binomial, Poisson and normal distribution was the buttons they pressed.
     
  13. Piranha

    Piranha Lead commenter

    One advantage of those using MEI - it is in C1 so they need to be able to do it without a calculator.
     
  14. When I do the topic at C2 I make a point of doing examples to the power of 5 on paper or mentally which many saying "Its on my calculator anyway"...saying that only 2 of 24 knew what that ! thingy meant
    At the end of the day, AS generally filters those who dragged themselves through AS on algorithms.
    I like to mix up methods used as early as possible to avoid them reaching for their copy of pascals triangle when the have a 1/2 power in C4
     
  15. None of us should underestimate the difficulty of the concepts and manipulations required in this general area. It took Mathematicians as great as Newton, Pascal etc. to get it off the ground, and even they had no real understanding of convergence.
    It takes time, so give the kids time.
    BTW, I often use Pascal's triangle - it takes about 10 seconds to write down the first 6 rows.

     
  16. I wouldnt for a minute expect a pupil to know the ins and outs, you are right, tht takes real power and years BUT i do expect a pupil to (i) know what a factorial is and (ii) be able to use either pencil and paper or their brain to find values such as 8C4.
    Do you feel thats a fair thing to ask from an A level pupil? (I may be out of touch)
     
  17. I was really thinking in terms of what the OP was asking about.
    Of course hand calculations are important, particularly how to divide by common factors, and then that leads on to the more general binomial theorem stuff.
    The 8Crs are good practice for the 7 times table.
    Factorials can of course be introduced to almost any yeargroup - see how quickly they grow.
     
  18. Agree Polecat, thankyou
     

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