Find the limit of the sequence $(b_n)_{n=1}^{\infty}$ defined by
$$b_{n} = \sum_{k=1}^{n} \frac{n}{n^2+k} = \frac{n}{n^2+1} + \ldots +\frac{n}{n^2+n} $$
for $n \geq 1$. I know that I have to use Squeezing Principle but I do not know how to find the lower and upper bound
If you enjoy harmonic numbers $$b_n=\sum_{k=1}^{n} \frac{n}{n^2+k}=n \left(H_{n^2+n}-H_{n^2}\right)$$ Using the asymptotics $$H_p=\gamma +\log \left({p}\right)+\frac{1}{2 p}-\frac{1}{12 p^2}+O\left(\frac{1}{p^4}\right)$$ apply it twice an continue with Taylor expansion to get $$b_n=1-\frac{1}{2 n}-\frac{1}{6 n^2}+\frac{1}{4 n^3}+O\left(\frac{1}{n^4}\right)$$
If you compute exactly $b_{10}=\frac{11210403701434961}{11818204429243212}\approx 0.948571$ while the above truncated formula would give $\frac{11383}{12000}\approx 0.948583$
