Results 1 to 2 of 2

Thread: A limit

  1. #1
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14

    A limit

    For any integrable function on $\displaystyle [0,1]$ compute the value of $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f(x)\,dx}}.$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14
    Write $\displaystyle \sum\limits_{k=0}^{n-1}{(n-k)\int_{\frac{k}{n}}^{\frac{k+1}{n}}{f}}=\sum\limi ts_{k=0}^{n-1}{\sum\limits_{j=k+1}^{n}{\int_{\frac{k}{n}}^{\fr ac{k+1}{n}}{f}}}$ and reverse summation order.

    --------

    After reversing summation order we have,

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

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

Similar Math Help Forum Discussions

  1. Replies: 12
    Last Post: Aug 26th 2010, 10:59 AM
  2. Replies: 1
    Last Post: Aug 8th 2010, 11:29 AM
  3. Replies: 1
    Last Post: Feb 5th 2010, 03:33 AM
  4. Replies: 16
    Last Post: Nov 15th 2009, 04:18 PM
  5. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Sep 3rd 2009, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum