1. Determine whether each of these series converge absolutely, converge conditionally, or diverge. \[ \sum_{n=1}^\infty (-1)^{n+1}\frac{n^2}{n^4+3} \qquad \qquad \sum_{n=1}^\infty (-1)^{n-1}\frac{3n}{4n-1} \qquad \qquad \sum_{n=1}^\infty \frac{\sin(n)}{n^4} \] \[ \sum_{n=42}^\infty \frac{(-1)^{n}}{n^7} \qquad \qquad \sum_{n=1}^\infty \frac{(-1)^{n}}{\sqrt[7]{n}} \qquad \qquad \sum_{n=1}^\infty (-1)^{n}\frac{5n}{3n+7} \] \[ \sum_{n=1}^\infty \frac{(-1)^{n+4}}{\ln\!\left(n^2\right)} \qquad \qquad \qquad \sum_{n=7}^\infty (-1)^{n}\left(\sqrt{n+7} - \sqrt{n} \right) \]
2. For what values of \(p\) is this series convergent? \[\sum_{n=2}^\infty (-1)^{n-1}\frac{\big(\ln(n)\big)^p}{n}\]