If you consider the sum $s_n = n / n^2 + n/ (n^2 + 1) + ... + n / (n^2 + (n - 1)^2)$ and give an upper and a lower estimate for s$_n$ in terms of the integral from 0 to n $I_n = ∫ n/(n^2 + x^2))dx$. How can you use this to compute the limit of $s_n$ as $n → ∞$?
This is what I’ve tried so far to solve this question: https://i.stack.imgur.com/OqQE0.jpg
But as you can see I got stuck. Any feedback, help or tips would be very much appreciated! Thank you in advance!
$\frac n {n^{2}+(j+1)^{2}} \leq \int_j^{j+1} \frac n {n^{2}+x^{2}}dx \leq \frac n {n^{2}+j^{2}}$. Summing from $j=1$ to $j=n-1$ we find that $s_n-\frac 1 {2n} \leq \int_1^{n} \frac n {n^{2}+x^{2}} \leq s_n$. Also $\int_j^{j+1} \frac n {n^{2}+x^{2}}dx=\arctan (\frac x n) |_1^{n}$ from which you can see that $s_n \to \arctan 1=\frac {\pi} 4$.