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Math Help - Area?

  1. #1
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    Area?

    Find a formula for for sum of n terms.Use the formula to find limit as n approaches infinity.

    sigma notation start i=1 end n

    ((1+(i/n))((2/n))

    Would anyone kindly help me with this problem.
    Last edited by homeylova223; August 9th 2011 at 09:29 AM. Reason: Solved
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Area?

    I guess you mean this:
    \sum_{i=1}^{n} \frac{1+\frac{i}{n}}{\frac{2}{n}}
    ?
    Write it as:
    \sum_{i=1}^{n} \frac{1+\frac{i}{n}}{\frac{2}{n}}=\sum_{i=1}^{n} \frac{\frac{n+i}{n}}{\frac{2}{n}}=\sum_{i=1}^{n} \frac{n+i}{2}

    Do you recognize a 'special' serie? ...
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Area?

    Quote Originally Posted by homeylova223 View Post
    Find a formula for for sum of n terms.Use the formula to find limit as n approaches infinity.

    sigma notation start i=1 end n

    ((1+(i/n))((2/n))

    Would anyone kindly help me with this problem.

    It is Riemann's upper sum of function f(x)=1+x/2 on [0,2].

    \int_0^2 (1+x/2) \ dx =3
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  4. #4
    Grand Panjandrum
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    Re: Area?

    Quote Originally Posted by homeylova223 View Post
    Find a formula for for sum of n terms.Use the formula to find limit as n approaches infinity.

    sigma notation start i=1 end n

    ((1+(i/n))((2/n))

    Would anyone kindly help me with this problem.
    This is a Riemann sum for:

    I=\int_0^2 (1+x/2)\; dx

    If you have arrived at the limit from the Riemann sum you should say so, otherwise you will probably get the integral as the answer for the limit rather than the sum of the finite series followed by limiting.

    CB
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  5. #5
    Grand Panjandrum
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    Re: Area?

    Quote Originally Posted by Siron View Post
    I guess you mean this:
    \sum_{i=1}^{n} \frac{1+\frac{i}{n}}{\frac{2}{n}}
    ?
    That should be:

    \sum_{i=1}^{n} \left(1+\frac{i}{n}\right)\frac{2}{n}}=2+\frac{2}{  n^2}\sum_{i=1}^n} i

    CB
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Area?

    Quote Originally Posted by CaptainBlack View Post
    That should be:

    \sum_{i=1}^{n} \left(1+\frac{i}{n}\right)\frac{2}{n}}=2+\frac{2}{  n^2}\sum_{i=1}^n} i

    CB
    Yes, offcourse, I red it wrong.
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  7. #7
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    Re: Area?

    I do not know how to explain I am trying to find the sum of the finite series. But to do that would I apply the interval Zartathrusta and you gave me?
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  8. #8
    MHF Contributor Siron's Avatar
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    Re: Area?

    Look at the post 5 of Captain Black. Calculate \lim_{n \to \infty} of the sum that will give you the area.
    Afterwards compare with post 3 (the integral) of Also sprach zarathrusta, you'll come to the same result.
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