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Math Help - Limit

  1. #1
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    Limit

    IF f(x)=\frac{x}{\sqrt{x+1}},
    Calculate
    \underset{n \to \infty }{lim}\left [f\left (  \frac{1}{n^2}\right ) +f\left (  \frac{2}{n^2}\right ) + ... +f\left (  \frac{n}{n^2}\right )  \right ]
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  2. #2
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    Quote Originally Posted by mp3qz View Post
    IF f(x)=\frac{x}{\sqrt{x+1}},
    Calculate
    \underset{n \to \infty }{lim}\left [f\left ( \frac{1}{n^2}\right ) +f\left ( \frac{2}{n^2}\right ) + ... +f\left ( \frac{n}{n^2}\right ) \right ]
    Try and relate the limit to a Riemann sum.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Try and relate the limit to a Riemann sum.

    if I have misspelled, hope you sympathize
    I don't really undertand
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  4. #4
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    Quote Originally Posted by mp3qz View Post
    if I have misspelled, hope you sympathize
    I don't really undertand
    Have you learned about Riemann sums?
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  5. #5
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    Here is an illustration.

    \displaystyle \underset{n \to \infty }{\lim}\left [f\left ( \frac{1}{n^2}\right ) +f\left ( \frac{2}{n^2}\right ) + ... +f\left ( \frac{n}{n^2}\right ) \right ] = \underset{n \to \infty }{\lim}\sum_{k=1}^n f\left ( \frac{k}{n^2}\right ) =

    [Math]\displaystyle = \underset{n \to \infty }{\lim}\sum_{k=1}^n \frac{k/n^2}{\sqrt{k/n^2+1}} [/tex]

    which I'm doubtful that you can relate to a Riemann sum, since it doesn't sum to n^2 but to n (so the interval of summation gets smaller and smaller every time we increase n) and we are missing the multiplication by the length of the subdivision 1/n^2. This is in fact why it converges because if one is present without the other we either get divergence or convergence to 0.
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