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Simple Math Question

  1. Dec 5, 2004 #1
    Greetings. Alright, if anyone's bored enough to be on-line right now, what is the following simplified and how do you get it?

    (1-1/2)(1-1/3)(1-1/4)(1-1/5)...(1-1/n)

    Thanks for any help,

    - Alisa
     
  2. jcsd
  3. Dec 5, 2004 #2

    Tide

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    Add the fractions inside each set of parentheses and see if there is a pattern! :-)
     
  4. Dec 5, 2004 #3
    I got infinity(1-2/n+2), where n= the denominator of the first of the two fractions being multiplied. That doesn't sound too solid...
     
  5. Dec 5, 2004 #4

    Tide

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    [tex]\left(1 - \frac {1}{2}\right) \left(1 - \frac {1}{3}\right) \left(1 - \frac {1}{4}\right) \cdot \cdot \cdot \left(1 - \frac {1}{n}\right) = \frac {1}{2} \cdot \frac {2}{3} \cdot \frac {3}{4} \cdot \cdot \cdot \frac {n-1}{n}[/tex]

    Do you see a pattern yet?
     
  6. Dec 6, 2004 #5
    I see the pattern, but I still keep on getting weird-looking answers such as

    ∞!
    ---------
    ((∞-1)!+1)

    The (----) being a division sign. If the symbol doesn't come out, it's supposed to be infinity.
     
  7. Dec 6, 2004 #6

    Tide

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    Well, first off, your original post said nothing about extending it to infinity. But since that seems to be where you are headed consider that

    [tex]\frac {n-1}{n} = 1 - \frac {1}{n}[/tex]

    Now let n go to infinity! :-)
     
  8. Dec 6, 2004 #7
    Wait up, it's 2 A.M., and I can't think very straight. Why does (n-1)/n=1-(1/n)? And shouldn't we be doing factorials like ((n-1)!)/n! or something like that since all of this has to be multiplied?
     
  9. Dec 6, 2004 #8

    Tide

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    Um ... it's a fundamental property of numbers? The distributive property.

    You can certainly use factorials but why would you want to when all the intermediate factors cancel out?
     
  10. Dec 6, 2004 #9
    Ooh sorry haha I didn't realize what you were talking about. Okay, yeah, so (n-1)/n= 1-(1/n), I see that. So then wouldn't it be just 1/n if we consider all the factoring out?
     
  11. Dec 6, 2004 #10

    Tide

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    Exactly! I knew you'd see it sooner or later. :-)
     
  12. Dec 6, 2004 #11
    Haha thank you! Maybe next time I should try to get started a bit earlier...
     
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