Limit $\lim_{n\to \infty} n(\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\cdots+\frac{1}{(2n)^2})$ I need some help finding the limit of the following sequence:
$$\lim_{n\to \infty}  a_n=n\left(\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\cdots+\frac{1}{(2n)^2}\right)$$
I can tell it is bounded by $\frac{1}{4}$ from below and decreasing from some point which tells me it is convergent.
I can't get anything else though.
So far we did not have integrals and we just started series, so this should be solved without using either.
Some hints would be very welcome :)
Thanks

$$ \lim_{n\to\infty} n\sum_{k=n+1}^{2n} \frac{1}{k^2}=\lim_{n\to\infty} \frac{1}{n}\sum_{k=n+1}^{2n}\frac{1}{\left(\frac{k}{n}\right)^2}=\int_1^2 \frac{1}{x^2} \mathrm dx=\left[-\frac{1}{x}\right]_1^2=\frac{1}{2} $$
