<s>[INST] Limit of the sequence $\frac{1}{n^2+1}+\frac{2}{n^2+2}+...+\frac{n}{n^2+n}$ As in the title, we have to calculate the limit $$\frac{1}{n^2+1}+\frac{2}{n^2+2}+...+\frac{n}{n^2+n}$$ when $n\rightarrow+\infty$. It is easy to see that $$\lim_{n\to\infty} \frac{1}{n^2+1}+\frac{1}{n^2+2}+...+\frac{1}{n^2+n}=0$$ and $$\lim_{n\to\infty} \frac{n}{n^2+1}+\frac{n}{n^2+2}+...+\frac{n}{n^2+n}=1$$
I don't know how to calculate the limit. I am  pretty sure that we somehow apply the sandwich rule, but I am afraid that there is something similary with $$\lim_{n\to\infty} \frac{1}{1^2}+\frac{1}{2^2}+...+\frac{1}{n^2}=\frac{\pi^2}{6}$$    
 [/INST] Hint:
$$\lim_{n\to\infty} \frac{1}{n^2+1}+\frac{2}{n^2+1}+...+\frac{n}{n^2+1}=\frac{1}{2}$$ and
$$\lim_{n\to\infty} \frac{1}{n^2+n}+\frac{2}{n^2+n}+...+\frac{n}{n^2+n}=\frac{1}{2}$$
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