# Sigma notation help

• Nov 29th 2009, 05:08 PM
shmayer
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.
• Nov 29th 2009, 07:07 PM
Jameson
Quote:

Originally Posted by shmayer
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.

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:

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