This is question 1a in section 6.3.6 of Zorich's Analysis I.
I went about it as follows:
$ \lim_{n\to\infty}\left[ \frac{n}{(n+1)^2} + ... + \frac{n}{(2n)^2}\right] = \lim_{n\to\infty}\left[ \frac{n}{[n(1+1/n)]^2} + ... + \frac{n}{[n(1+n/n)]^2}\right] = \lim_{n\to\infty}\left[ \frac{n}{n^2}\frac{1}{(1+1/n)^2} + ... + \frac{n}{n^2}\frac{1}{(1+n/n)^2}\right] = \lim_{n\to\infty}\left[ \frac{1}{n}\frac{1}{(1+1/n)^2} + ... + \frac{1}{n}\frac{1}{(1+n/n)^2}\right] $
This is the (upper) Riemman sum for the function $f(x) = 1/x^2$ on [1, 2] and therefore is equal to:
$ \int_{1}^{2}\frac{1}{x^2}dx = -\frac{1}{2} - (-1) = \frac{1}{2}$
A quick numeric check with Mathematica seems to confirm:
In:= Limit[Sum[n/((n + i)^2), {i, 1, n}], n -> Infinity]
Out= 1/2
Is this right?