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Math Help - A limit

  1. #1
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    A limit

    For any integrable function on [0,1] compute the value of \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f(x)\,dx}}.
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  2. #2
    Math Engineering Student
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    Write \sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f}}=\sum\limi  ts_{k=0}^{n-1}{\sum\limits_{j=k+1}^{n}{\int_{\frac{k}{n}}^{\fr  ac{k+1}{n}}{f}}} and reverse summation order.

    --------

    After reversing summation order we have,

    \sum\limits_{k=0}^{n-1}{\sum\limits_{j=k+1}^{n}{\int_{\frac{k}{n}}^{\fr  ac{k+1}{n}}{f}}}=\sum\limits_{j=1}^{n}{\sum\limits  _{k=0}^{j-1}{\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f}}}=\sum\li  mits_{j=1}^{n}{\int_{0}^{\frac{j}{n}}{f}}.

    Thus, the limit becomes \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{j=1}^{n}{\int_{0}^{\fr  ac{j}{n}}{f}}=\int_0^1\int_0^yf(x)\,dx\,dy, and the rest follows by reversing the integration order. \blacksquare
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