<s>[INST] Problem with counting limit. I am obliged to count the
$$\lim_{n\rightarrow\infty} a_n$$
where
$$a_n=\frac{1}{n^2+1}+\frac{2}{n^2+2}+\frac{3}{n^2+3}+\cdots+\frac{n}{n^2+n}.$$
I know that this type of task, I should use squeeze theorem. I should find two sequences one that is smaller and one that is bigger. So I find $\large b_n=\frac{1}{n^2+1}$, $ b_n \le a_n$, but I have trouble in finding the bigger sequence $c_n \ge a_n$. I will be glad for any help.
 [/INST] Each term in the expression for $a_n$ is less than or equal to $\displaystyle \frac{n}{n^2+n} = \frac{1}{n+1}$, so I think the upper bound would be $$a_n = \frac{1}{n^2+1} + \frac{2}{n^2+2} + \cdots + \frac{n}{n^2+n} \le \frac{n}{n^2+n} + \frac{n}{n^2+n} + \cdots + \frac{n}{n^2+n} = \frac{n}{n+1} = c_n.$$
Of course, as $n \rightarrow \infty$, $c_n \rightarrow 1$.
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