Example 7. Investigate the convergence of the series $\sum_{n=1}^{\infty}\left(\frac{n+1}{3 n-1}\right)^{n^{2}}$.
Solution. We apply the Cauchy radical test. We have:

$$
q=\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=\lim _{n \rightarrow \infty}\left(\frac{n+1}{3 n-1}\right)^{n}=\lim _{n \rightarrow \infty}\left(\frac{1+\frac{1}{n}}{3-\frac{1}{n}}\right)^{n}=0
$$

Answer. The series converges.