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Math Help - Simple limit escaping me

  1. #1
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    Simple limit escaping me

    What is

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

    ??

    I guessed that the answer would be \log(2), and simulation confirms it, but how do I prove it?
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  2. #2
    Super Member girdav's Avatar
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    Re: Simple limit escaping me

    By Riemann sums: \sum_{k=1}^n\frac 1{k+n}=\frac 1n\sum_{k=1}^n}\frac 1{\frac kn+1}=\frac 1n\sum_{k=1}^nf\left(\frac kn\right), where f(x)=\frac 1{x+1}. Since \int_0^1f(t)dt=\log 2, you are done.
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