$$X := \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2}+...+\frac{1}{(n+n)^2}$$ Show the above sequence converges to $0$
I was able to prove the above sequence converges : $$\sum_{n=1}^\infty \frac{1}{4n^2} < \sum_{n=1}^\infty \frac{1}{n^2}$$ thus by comparison test the above sequence converges. $$\\$$But I wasn't able to prove it converges to $0$? Solution or hints would be appreciated!
$$ \leq n\cdot \frac{1}{(n+1)^2} < \frac{n}{n^2} = \frac{1}{n} \to 0 \text{ as } n\to\infty. $$
– Adam Rubinson Oct 30 '23 at 14:32