Strategies for Computing Limits Involving Square Roots and Rational Expressions

[Caldus]

I'm trying to isolate the i variable in this formula. Thanks for any help.

\lim_{n->infinity}\frac{5}{n}\sum_{i=1}^{n}\sqrt{25-\frac{5i^2}{n^2}}

 

[mathman]

i is a dummy variable for the summation. What do you mean by "isolate"? Also by inspection, it looks like the limit can be gotten by integral from 0 to 1 of 5(25-5x²)^1/2.

 

[Caldus]

I'm trying to get i to be by itself inside the summation sign but I don't know how to get rid of that square root.

 

[humanino]

I don't understand the last post. But if you want only to calculate the limit, mathman gave you the right direction (integrate)

 

[Hurkyl]

You won't be able to isolate i in this equation.

To compute that limit directly, I think you'll pretty much have to do some sort of clever approximation, and prove the error in the approximation goes to zero as n goes to infinity. Actually carrying out this programme is well beyond what would be expected in a calc II Class.

This limit presumably comes from some sort of integral, probably \int_0^5 \sqrt{5^2 - x^2} \, dx. There's a clever choice of partition for which the Riemann sum is much easier to compute, but as mathman was trying to hint, there's a much easier way to come up with the value of this integral...