Math Real Analysis Problem, Riemann Sum Integral?

[thunderx7]

Part 1. Homework Statement

The problem literally states...

"
The Integral.
limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2
"

According to the teacher, the answer is 1/2. I don't know why or how to get there.


Part 2. The attempt at a solution

I was able to get this far.

limit of n->infinity of the sum from k=n+1 to 2n of n/(k^2)

I put together a summation from whatever info I had, and somehow, that equals 1/2.

I think you are supposed to use Riemann Sum Integral or something like that.

Please explain or show me how with as much detail as possible if you can.

Thank you very much. This is my first post so sorry if it is rude or something is not done right. I read the sticky though! Sorry for the hassles.

 

[snipez90]

$$n\cdot\left(\frac{1}{(n+1)^2} + ... + \frac{1}{(2n)^2}\right) = \frac{1}{n}\cdot\left[ \left(\frac{n}{n+1}\right)^2 + ... + \left(\frac{n}{n + n}\right)^2\right]$$

This almost obviously looks like a Riemann sum (over what partition?), but you need to make one further manipulation to really see the corresponding function for which we're essentially taking the definite integral of.