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Math Help - Sigma notation help

  1. #1
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    Sigma notation help

    Find lim as n->infinity of 1/n(square root of 1/n + square root of 2/n + ....+ square root of n/n)

    I can get to 1/n lim as n->infinity i^1/2...i dont know where to go from here. Is there anyway i can isolate so i just have 'i' inside the notation and solve from there? Thanks.
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  2. #2
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    Quote Originally Posted by shmayer View Post
    Find lim as n->infinity of 1/n(square root of 1/n + square root of 2/n + ....+ square root of n/n)

    I can get to 1/n lim as n->infinity i^1/2...i dont know where to go from here. Is there anyway i can isolate so i just have 'i' inside the notation and solve from there? Thanks.
    I'm not sure about this response, but am 80% confident this is true.

    If you split the region of integration for f(x) into n rectangles, starting from 0 and ending at 1, the area is \lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(x_i) \Delta x. Now i is somewhere inbetween the beginning of the rectangle and the end, but as n goes to infinity any method of finding i will work.

    So let x=\frac{i}{n} \text{ and} dx=\frac{1}{n} Then I believe this leads to the conclusion that your limit is the same as:

    \int_{0}^{1}\sqrt{x}dx
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