## Calculus (3rd Edition)

The sequence $r_n$ diverges.
Since we have $$\lim_{n\to \infty}r_n=\lim_{n\to \infty} \ln n -\ln (n^2+1) =\lim_{n\to \infty}\ln\left(\frac{n}{n^2+1}\right)\\ =\ln \lim_{n\to \infty}\frac{1/n}{1+(1/n^2)}=\ln 0=-\infty,$$ then the sequence $r_n$ diverges.