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Summation of a Logarithmic Series

  1. Jul 10, 2012 #1

    S.R

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    1. The problem statement, all variables and given/known data
    What is the sum of the following series?

    log(3/2)+log(4/3)+log(5/4)+...log(200/199).

    Where log(x) is log base 10 of x.

    2. Relevant equations



    3. The attempt at a solution
    Evidently, the previous form equals:

    log(3/2*4/3*5/4*...200/199)

    I'm missing something - no patterns are evident to me other than the denominator and numerator of the subsequent terms cancel out.

    Any guidance would be appreciated.
     
  2. jcsd
  3. Jul 10, 2012 #2

    Curious3141

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    And that's important! What are you left with after cancelling *everything* that can be cancelled?
     
  4. Jul 10, 2012 #3
    [tex]log(\frac{3}{2}*\frac{4}{3}*\frac{5}{4}*\frac{6}{5}..........*\frac{200}{199})[/tex]

    Do you see in what pattern the terms cancel and which terms are left? :wink:
     
  5. Jul 10, 2012 #4

    S.R

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    Intuitively, yes - would it be correct in saying that all terms cancel other than 1/2 and 200/1, leaving 200/2?
     
  6. Jul 10, 2012 #5

    Mentallic

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    Yes, that would be correct, and you didn't find it intuitive even though you spotted the pattern?
     
  7. Jul 10, 2012 #6
    Yes, only 1/2 and 200/1 remains. 3 cancels 1/3, 4 cancels 1/4, 5 cancels 1/5 but there's no one to cancel 200 and 1/2.
     
  8. Jul 10, 2012 #7

    S.R

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    Quickly edited after rereading :frown:. I'm not sure why I didn't recognize that pattern before.
     
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