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Math Help - express limit as definite integral

  1. #1
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    express limit as definite integral

    Hi I think this would be easier if there was a range but one was not given. Here is the entire question:

    n
    Express Lim sum notation (i^5) / (n^6) as a definite integral.
    n -> infinity i=1

    Then evaluate the integral.

    my problem is there is no x in this formula and but I guess it can be f(i) instead of the usual f(x) but then what does that mean for n?

    Any insight is appreciated.

    this is my second course but I am just as lost.

    A calculus beginner
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  2. #2
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    Try \int_{0}^{1}x^{5}dx

    {\Delta}x=\frac{1-0}{n}=\frac{1}{n}

    i{\Delta}x=\frac{i}{n}

    f(x_{k})\cdot {\Delta}x=(\frac{i}{n})^{5}\cdot \frac{1}{n}=\frac{i^{5}}{n^{6}}

    Look anything like what you have?.

    \frac{1}{n^{6}}\sum_{i=1}^{n}i^{5}=\frac{(n+1)^{2}  (2n^{2}+2n-1)}{12n^{4}}=\frac{1}{2n}+\frac{5}{12n^{2}}-\frac{1}{12n^{4}}+\frac{1}{6}

    \lim_{n\to \infty}\left[\frac{1}{2n}+\frac{5}{12n^{2}}-\frac{1}{12n^{4}}+\frac{1}{6}\right]=\frac{1}{6}

    Check against \int_{0}^{1}x^{5}dx
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