1. Determine whether each of these series converge absolutely, converge conditionally, or diverge. \[ \sum_{n=4}^\infty \frac{1}{n!} \qquad \qquad \sum_{n=1}^\infty (-1)^{1-n}\frac{n^7}{7^n} \qquad \qquad \sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n \] \[ \sum_{n=1}^\infty \left(\frac{4n+8}{9n+7}\right)^n \qquad \qquad \sum_{n=1}^\infty \frac{n!}{n^n} \qquad \qquad \sum_{n=1}^\infty \frac{n^{69}69^n}{69n!} \] \[ \sum_{n=1}^\infty \left(\frac{\ln(n)}{n}\right)^n \qquad \qquad \sum_{n=1}^\infty n^\mathrm{e}\mathrm{e}^{-n} \qquad \qquad \sum_{n=1}^\infty \frac{(2n)!}{(n!)^2} \]
2. Show that regardless of the value of \(x\), the series \[\sum_{n=0}^\infty \frac{x^n}{n!} \] converges, and realize that this means \( \lim_{n\to\infty} x^n/n! = 0 \) for any number \(x\).