Answer
The sequence $r_n$ diverges.
Work Step by Step
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.
