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Math Help - My problem

  1. #1
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    My problem

    Calculate :
    P= (1-1/2)(1-1/3)(1-1/4)........(1-1/2006)(1-1/2007)
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  2. #2
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    Quote Originally Posted by Tenrasmey
    Calculate :
    P= (1-1/2)(1-1/3)(1-1/4)........(1-1/2006)(1-1/2007)
    This is what you got,
    \left( 1-\frac{1}{2} \right) \left(1-\frac{1}{3} \right)\cdot .... \cdot \left(1-\frac{1}{2007} \right)
    Combine your fractions,
    \left( \frac{1}{2} \right) \left(\frac{2}{3} \right) \left( \frac{3}{4} \right)\cdot ... \cdot \left( \frac{2005}{2006} \right) \left(\frac{2006}{2007} \right)
    Each denominator kills the numerator thus,
    (1)(1)....(1)\left(\frac{1}{2007} \right) =1/2007
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  3. #3
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    Quote Originally Posted by ThePerfectHacker
    This is what you got,
    \left( 1-\frac{1}{2} \right) \left(1-\frac{1}{3} \right)\cdot .... \cdot \left(1-\frac{1}{2007} \right)
    Combine your fractions,
    \left( \frac{1}{2} \right) \left(\frac{2}{3} \right) \left( \frac{3}{4} \right)\cdot ... \cdot \left( \frac{2005}{2006} \right) \left(\frac{2006}{2007} \right)
    Each denominator kills the numerator thus,
    (1)(1)....(1)\left(\frac{1}{2007} \right) =1/2007
    It is the product equivalent of a telescoping* series, in fact if you take
    logs it is a telescoping series.

    RonL




    *telescoping series: A series \sum_{r=1}^n a_r which can be
    written in the form:

    <br />
\sum_{r=1}^n a_r =\sum_{r=1}^n (b_{r}-b_{r+1})=b_1-b_{n+1}<br />

    for some sequence \{b_r,\ r=1,\dots n+1 \}

    RonL
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  4. #4
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    I just have a question about infinite product.

    If an infinite product is positive terms then to work with convergence what do you handle it? Do you take the log and turn it into an infinite sum and work with that?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    I just have a question about infinite product.

    If an infinite product is positive terms then to work with convergence what do you handle it? Do you take the log and turn it into an infinite sum and work with that?
    I can't say that I have had to deal with many products (finite or infinite),
    but taking logs is one of the first tricks I would normally employ, after all
    it is a quick method of deriving Stirling's approximation to the factorial.

    However there must be many cases where this approach will not be
    useful.

    RonL
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  6. #6
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    Quote Originally Posted by ThePerfectHacker
    I just have a question about infinite product.

    If an infinite product is positive terms then to work with convergence what do you handle it? Do you take the log and turn it into an infinite sum and work with that?
    I don't know about doing general problems that way, but it's done that way in Statistical Mechanics all the time. Of course, everyone knows that Physicists mangle Math methods all the time...

    -Dan
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