2.10.29 Number series

Problem: Examine the series for convergence: 1) \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(n+1)} \), 2) \( \sum_{n=1}^{\infty} \frac{n^{4}}{4^{n}+1} \), 3) \( \sum_{n=1}^{\infty}\left(\frac{n}{3 n-1}\right)^{2 n-1} \).