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Math Help - Find the limit

  1. #1
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    Find the limit

    Find the limit when n\rightarrow\infty of the series
    \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+....+\fr  ac{1}{2n}
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  2. #2
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    Quote Originally Posted by varunnayudu View Post
    Find the limit when n\rightarrow\infty of the series
    \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+....+\fr  ac{1}{2n}
    \lim_{n \to \infty} \sum_{k=1}^{k=n} \frac1{k+n} = \lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx = \ln 2

    \lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx is obtained by observing that the left hand side is a Riemann sum.
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