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Math Help - Show that limit is a definite integral

  1. #1
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    Show that limit is a definite integral

    Show that the given limit is a definite integral \int_a^b f(x) \, dx for a suitable interval [a,b] and function f


    The limit is:


    \lim \limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n}{n^2 + i^2}


    That's the problem I can't solve. Here is how far I've made it:


    Since a definite integral can be defined as


    \int_a^b f(x) \, dx = \lim\limits_{n \to \infty} \sum\limits_{i=1}^n f\left(a + (b-a)\frac{i}{n}\right) \cdot \frac{b-a}{n}


    Then:


    f\left(a + (b-a)\frac{i}{n}\right) \cdot \frac{b-a}{n} = \frac{n}{n^2 + i^2}


    which simplifies to:


    f\left(a + (b-a)\frac{i}{n}\right)= \frac{1}{b-a} \frac{n^2}{n^2 + i^2}


    I'm not sure how to solve, but, I can guess that b-a=1 which would simplify things to:


    f\left(a + \frac{i}{n}\right)= \frac{n^2}{n^2 + i^2}


    I can also guess that a=1 so that:


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


    Now, I can't find a function f that satisfies this. f(x)=\frac{1}{x^2} almost works but not quite.
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  2. #2
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    Re: Show that limit is a definite integral

    Quote Originally Posted by VinceW View Post
    Show that the given limit is a definite integral \int_a^b f(x) \, dx for a suitable interval [a,b] and function f
    The limit is:
    \lim \limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n}{n^2 + i^2}
    Let a=0,~b=1~\&~f(x)=\frac{1}{1+x^2}.
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  3. #3
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    Re: Show that limit is a definite integral

    that works. I take it that was just intuition? Not sure how I would have figured that out by myself. thanks!
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