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Math Help - Telescoping series

  1. #1
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    Telescoping series

    Dont quite understand how i would solve the following problem:

    Repersent the series as a telescopic series: \sum ln(n(n+2)/(n+1)^2) n=1 to infiniti
    Last edited by NFS1; August 15th 2012 at 07:50 AM.
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  2. #2
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    Re: Telescoping series

    First, read about telescoping series. Then prove that

    \sum_{n=1}^N\ln\frac{n(n+2)}{(n+1)^2}=\ln1-\ln2-\ln(N-1)+\ln N

    using properties of ln.

    Edit: The right-hand side should be \ln1-\ln2-\ln(N+1)+\ln(N+2).
    Last edited by emakarov; August 15th 2012 at 10:54 AM.
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  3. #3
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    Re: Telescoping series

    but what is the series represented as a partial fraction? I get:

    -1/(n+1)^2

    substituting in n=1 and so on i get ln-1/4, ln-1/9...which surely cannot be right
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  4. #4
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    Re: Telescoping series

    Quote Originally Posted by NFS1 View Post
    but what is the series represented as a partial fraction?
    Hmm, I could not parse this sentence. Edit: OK. I parsed it, but I am not sure how to represent this series as a partial fraction.

    Quote Originally Posted by NFS1 View Post
    I get:

    -1/(n+1)^2
    What do you claim about this expression? Obviously, \frac{n(n+2)}{(n+1)^2}\ne-\frac{1}{(n+1)^2}.

    I am not sure partial fractions are necessary here. Simplify the expression under the sum using the formulas for logarithm of product and quotient.
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  5. #5
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    Re: Telescoping series

    Quote Originally Posted by emakarov View Post
    Hmm, I could not parse this sentence. Edit: OK. I parsed it, but I am not sure how to represent this series as a partial fraction.

    What do you claim about this expression? Obviously, \frac{n(n+2)}{(n+1)^2}\ne-\frac{1}{(n+1)^2}.

    I am not sure partial fractions are necessary here. Simplify the expression under the sum using the formulas for logarithm of product and quotient.
    Don't quite understand how to do that. I thought to express the series as telescopic series you should express the series as partial sums. If you can show a step by step solution it will be greatly appreciated.
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  6. #6
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    Re: Telescoping series

    The point is that ln\left(\frac{n(n+2)}{(n+1)^2}\right)= ln(n+2)+ ln(n)- 2ln(n+1)= \left(ln(n+2)- ln(n+1)\right)- \left(ln(n+1)- ln(n)\right)
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  7. #7
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    Re: Telescoping series

    Quote Originally Posted by HallsofIvy View Post
    The point is that ln\left(\frac{n(n+2)}{(n+1)^2}\right)= ln(n+2)+ ln(n)- 2ln(n+1)= \left(ln(n+2)- ln(n+1)\right)- \left(ln(n+1)- ln(n)\right)
    Yes, this is the best representation.

    I was doing it as follows.

    \sum_{n=1}^N\ln\left(\frac{n(n+2)}{(n+1)^2}\right)  =
    \sum_{n=1}^N(\ln n+\ln(n+2)-2\ln(n+1))=
    (\ln1+\ln3-2\ln2)+(\ln2+\ln4-2\ln3)+(\ln3+\ln5-2\ln4)+ ... +(\ln(N-1)+\ln(N+1)-2\ln N)+(\ln N+ \ln(N+2)-2\ln(N+1))=
    \ln1-\ln2-\ln(N+1)+\ln(N+2)

    Attachment 24517

    The horizontal axis corresponds to arguments of ln and the vertical axis corresponds to terms of the series. Blue dots (k, n) denote ln(k) in the nth term, and red dots (k, n) denote -2ln(k) in the nth term. Then three dots on the same vertical line correspond to ln(k) - 2ln(k) + ln(k) = 0, so only dots surrounded by the dashed line remain.

    This all means that I was wrong about the partial sum in post #2.
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  8. #8
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    Re: Telescoping series

    Ok so the series represented as telescopic series is ln1 - ln2 + ln(n+2) - ln (n+1).

    How would i go about finding the sum of the telescopic series? Would i use log laws again so that the telescopic series is ln1/2 + ln(n+2)/(n+1)?? If so what would the answer to the sum be??
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  9. #9
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    Re: Telescoping series

    Now that you have what the sum is up to n, take the limit as n goes to infinity.
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  10. #10
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    Re: Telescoping series

    Quote Originally Posted by Prove It View Post
    Now that you have what the sum is up to n, take the limit as n goes to infinity.
    What is the sum?? is it: ln1 - ln2 + ln(n+2) - ln (n+1) or using log laws is it ln(1/2 + (n+2)/(n+1))?
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  11. #11
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    Re: Telescoping series

    Quote Originally Posted by NFS1 View Post
    What is the sum?? is it: ln1 - ln2 + ln(n+2) - ln (n+1) or using log laws is it ln(1/2 + (n+2)/(n+1))?
    The first is correct, but the second is not. It should be \displaystyle \begin{align*} \ln{\left[ \frac{1(n+2)}{2(n+1)} \right]} \end{align*}, and now take the limit.
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