<s>[INST] Sum of a series when $n$ goes to infinity How can we determine the sum
$$\lim_{n \to \infty} \left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots+\frac{n}{n^2+n}\right)$$
I tried to reduce this to an integral problem by dividing both numerator and denominator by $n^2$ but we get the term $\dfrac{\frac{1}{n}}{1+\frac{r}{n^2}}$ where $r$ ranges from $1$ to $n$ but to get the variable $x$ in integration we would need $\frac{r^2}{n^2}=x^2$. So,i am not sure how to do this. I will be very grateful for the solutions.
 [/INST] We have
$$\frac{n^2}{n^2+n} \leq\left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots+\frac{n}{n^2+n}\right) \leq \frac{n^2}{n^2+1},$$
so by the squeeze theorem
$$\lim_{n \to \infty} \left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots+\frac{n}{n^2+n}\right)=1.$$
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