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Math Help - How to do a Summation

  1. #1
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    How to do a Summation

    Hey everyone, these are 3 of like 15 problems that I need help with. I don't expect someone to do all three, but to pick one they are confident with and try to help me. If you can, thanks =D

    1)

    Evaluate (Find the sum of):



    Thanks
    Last edited by topsquark; May 14th 2011 at 10:45 AM. Reason: Moving questions that have no bearing to each other
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  2. #2
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    Can you simplify this sum?
    \sum\limits_{k = 0}^3 {\ln \left( {\frac{{k + 1}}{{k + 3}}} \right)}  = \left( {\ln (1) - \ln (3)} \right) + \left( {\ln (2) - \ln (4)} \right) + \left( {\ln (3) - \ln (5)} \right) + \left( {\ln (4) - \ln (6)} \right) = ?

    If so look for the pattern.
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  3. #3
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    Hello, PapaSmurf!

    \text{Evaluate: }\:S \;=\;\sum^{125}_{k=0} \ln\left(\frac{k+1}{k+3}\right)

    We have: . S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128})

    . . S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right)

    . . . . =\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right)

    . . . . =\;-\ln8128 \;=\;-9.00307017

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  4. #4
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    Quote Originally Posted by Soroban View Post
    Hello, PapaSmurf!


    We have: . S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128})

    . . S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right)

    . . . . =\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right)

    . . . . =\;-\ln8128 \;=\;-9.00307017

    WOW, THANK YOU SO MUCH!! Dude I had no idea that you could do that with 'ln's, I completely forget about it after finals in January

    Thanks again! How do I give rep to you?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by PapaSmurf View Post
    How do I give rep to you?
    You have two choices. First, "thanking" a member does give reputation. If you insist on rep, though, you cannot give rep after thanking a member. So what you will have to do is remove your thanks, then click on the six pointed star at the bottom left of the post.

    -Dan
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