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How to prove that this series bounded and monotonic

  1. Dec 10, 2008 #1
    Xn=(1-1/2)(1-1/4)..(1-(1/(2^n))

    i tried to prove that its monotonic
    by :
    1-1/(2^n) = (2^n-1)/2^n

    2^n -1 <2^n
    obviously its correct
    the numerator of each object is smaller then the denominator.

    what now??

    and how to prove that its bounded?
     
  2. jcsd
  3. Dec 10, 2008 #2

    Office_Shredder

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    You need to show that [tex]x_n > x_{n+1}[/tex] for all n given that [tex]x_n(1-2^{-n}) = x_{n+1}[/tex]

    It should be fairly easy from here
     
  4. Dec 10, 2008 #3
    that proves that its monotonic
    how to prove that its bounded?
     
  5. Dec 10, 2008 #4

    Office_Shredder

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    Is it increasing or decreasing?
     
  6. Dec 10, 2008 #5
    each next member is bigger then the previous one
    so its increasing

    1-1/2 1-1/4 1-1/8 etc..
     
  7. Dec 10, 2008 #6

    Office_Shredder

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    That's now what the sequence is. The sequence is

    1/2, 1/2*3/4, 1/2*3/4*7/8 etc.

    you should be able to see this from how xn is defined.
     
  8. Dec 10, 2008 #7
    ok so it getting smaller and smaller
    how to prove that its bounded?
     
  9. Dec 10, 2008 #8

    HallsofIvy

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    There's a pretty obvious lower bound. And since it is decreasing, isn't x1= 1/2 an upper bound?
     
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