<s>[INST] Computing $\lim_{n \to \infty} \left(\frac{n}{(n+1)^2}+\frac{n}{(n+2)^2}+\cdots+\frac{n}{(n+n)^2}\right)$ 
Calculate $$\lim_{n \to \infty} \left(\dfrac{n}{(n+1)^2}+\dfrac{n}{(n+2)^2}+\cdots+\dfrac{n}{(n+n)^2}\right).$$

I tried turning this into a Riemann sum, but didn't see how since we get $\dfrac{1}{n} \cdot \dfrac{n^2}{(n+k)^2}$, which I don't see how relates.
 [/INST] $\frac{n}{n+2}-\frac{n}{2n+1}=n\int\limits_{n+2}^{2n+1}\frac{1}{x^2}dx<n(\frac{1}{n+1}^2+\frac{1}{n+2}^2+\dots + \frac{1}{2n}^2)< n\int\limits_{n+1}^{2n}\frac{1}{x^2}dx=\frac{n}{n+1}-\frac{n}{2n}$
Clearly the expressions on both sides go to $\frac{1}{2}$ when $n\to \infty$.
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