<s>[INST] Find $\lim\limits_{n\to\infty} ( \frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\dots+ \frac{1}{(2n)^2})$ Find $\lim\limits_{n\to\infty} ( \frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\dots+ \frac{1}{(2n)^2})$
Please help me find this limit .
I cannot use cauchys limit theorem on this problem . So I am stuck and clueless
 [/INST] For $j\in \mathbb N$ and $x\in (j,j+1)$ we have $\frac {1}{(j+1)^2}<\frac {1}{x^2}$.  Therefore $$0<\sum_{j=n}^{2n-1}\frac {1}{(j+1)^2}=\sum_{j=n}^{2n-1}\int_j^{j+1}\frac {1}{(j+1)^2}dx<\sum_{j=n}^{2n-1}\int_j^{j+1}
\frac {1}{x^2}dx=$$ $$=\int_n^{2n}\frac {1}{x^2}dx=\frac {1}{n}- \frac {1}{2n}=\frac {1}{2n}.$$
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