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Math Help - What is the limit of this series?

  1. #1
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    What is the limit of this series?



    where a, b and e are constants. g and n are obvious from the sum. How can I find the limit of this series as n approaches infinity? I already tried several approaches, but it's always problematic as the "g" is under the square root sign.
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  2. #2
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    Re: What is the limit of this series?

    Quote Originally Posted by Supernova008 View Post


    where a, b and e are constants. g and n are obvious from the sum. How can I find the limit of this series as n approaches infinity? I already tried several approaches, but it's always problematic as the "g" is under the square root sign.
    First lump your constants, put A=|ea|, and c=|b/ea| then you have:

    A \sum_{k=1}^n \frac{1}{n}\sqrt{(k/n)^2+c^2}

    The sum can be interpreted as a Riemann sum for an integral from 0 to 1 of a continuous function so the limit is:

     A \int_0^1 \sqrt{x^2+c^2}\; dx

    CB
    Last edited by CaptainBlack; August 3rd 2011 at 06:09 AM.
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    Re: What is the limit of this series?

    Quote Originally Posted by CaptainBlack View Post
    First lump your constants, put A=|ea|, and c=|b/ea| then you have:

    A \sum_{k=1}^n \frac{1}{n}\sqrt{(k/n)^2+c^2}

    The sum can be interpreted as a Riemann sum for an integral from 0 to 1 of a continuous function so the limit is:

     A \int_0^1 \sqrt{x^2+c^2}\; dx

    CB
    I am not a university student yet, so pardon my lack of such advanced mathematical knowledge. I am currently researching this riemann sum, but could you tell me why this is so?

    EDIT: the first term of the sum is (1/n)*sqrt(1/n^2 + m^2)
    however, the first term of the riemann sum is 1/n*sqrt(n^2+m^2), right? isn't that an inconsistency?
    Last edited by Supernova008; August 3rd 2011 at 07:23 AM.
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  4. #4
    Grand Panjandrum
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    Re: What is the limit of this series?

    Quote Originally Posted by Supernova008 View Post
    I am not a university student yet, so pardon my lack of such advanced mathematical knowledge. I am currently researching this riemann sum, but could you tell me why this is so?

    EDIT: the first term of the sum is (1/n)*sqrt(1/n^2 + m^2)
    however, the first term of the riemann sum is 1/n*sqrt(n^2+m^2) , right? isn't that an inconsistency?
    What does the red text above refer to?


    The partitioning points are 1/n,\ 2/n,\ 3/n,\ ...\ ,\ n/n and h=1/n.

    (not advanced mathematical knowlege, this was taught in what would now be years 11 or 12 when I was in secondary education)

    CB
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  5. #5
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    Re: What is the limit of this series?

    Quote Originally Posted by CaptainBlack View Post
    What does the red text above refer to?


    The partitioning points are 1/n,\ 2/n,\ 3/n,\ ...\ ,\ n/n and h=1/n.

    (not advanced mathematical knowlege, this was taught in what would now be years 11 or 12 when I was in secondary education)

    CB
    Yeah sorry, I got it already, thank you very much!
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