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Math Help - proving limit of sum of series

  1. #1
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    proving limit of sum of series

    Show that
    \lim_{n\to\infty}\left( \sum_{k=1}^{n} \frac{n}{n^2+k^2}\right ) = \frac{\pi}{4}

    Can we use integral test to prove this? How do we go about doing this?
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  2. #2
    Member sbhatnagar's Avatar
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    Re: proving limit of sum of series

    Hello alphabeta89! Do you that  \int_{0}^{1} f(x) dx=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{1}{n}{f \Big(\dfrac{k}{n} \Big)

     \lim_{n \to \infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}=\lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{n}\frac{1}{1+\frac{k  ^2}{n^2}}

     =\int_{0}^{1}\frac{1}{1+x^2}dx=\Big( \arctan x\Big) \Big|_{0}^{1}

     =\frac{\pi}{4}
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