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Math Help - Finding the sum of an infinite sequence

  1. #1
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    Finding the sum of an infinite sequence

    Hey all, hope you're doing well. Here's the bonus question that's been bugging me for a while now:


    For anyone who can point me on the right track, your help is much appreciated.

    Thanks in advance!
    - Dave

    (NOTE: I know the answer is ln(2) from Maple; I just don't know how to get a ln from that sum)
    Last edited by urnidiot; January 29th 2009 at 09:41 PM.
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  2. #2
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    Quote Originally Posted by urnidiot View Post
    Hey all, hope you're doing well. Here's the bonus question's that's been bugging me for a while now:


    For anyone who can point me on the right track, your help is much appreciated.

    Thanks in advance!
    - Dave

    (NOTE: I know the answer is ln(2) from Maple; I just don't know how to get a ln from that sum)
    You should know the definition of a Riemann sum for this:
    \lim_{n \to \infty} \sum_{k=1}^{k=n} \frac1{n+k} = \lim_{n \to \infty}\frac1{n} \sum_{k=1}^{k=n} \dfrac1{1+\dfrac{k}{n}} = \int_{0}^{1} \frac1{1+x} \, dx
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    Thanks for the reply! However, I'm still not sure how to get from the second equation to the third - if you could elaborate that a little, I'd be very grateful.

    -Dave
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    Quote Originally Posted by urnidiot View Post
    Thanks for the reply! However, I'm still not sure how to get from the second equation to the third - if you could elaborate that a little, I'd be very grateful.

    -Dave
    Well, \int_0^1 \frac{dx}{x+1} =_{t=x+1} \int_1^2 \frac{dt}{t} = \ln 2.
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  5. #5
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    Thanks man, but that's not the part I'm having trouble understanding; it's the step before that to get to the integral.

    -Dave
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  6. #6
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    Quote Originally Posted by urnidiot View Post
    Thanks man, but that's not the part I'm having trouble understanding; it's the step before that to get to the integral.

    -Dave
    As I said, you should know the definition of a Riemann sum to understand that. Do you know what a Riemann sum is?

    I have used the following result:
    \lim_{n \to \infty} \frac1{n} \sum_{k=0}^{k=n} f\left(a + \frac{k(b-a)}{n}\right) = \int_a ^b f(x) \, dx
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