1. Determine whether each of these series converge or diverge. \[ \sum_{n=1}^\infty \frac{7}{3n^2+8n} \qquad \qquad \sum_{n=1}^\infty \frac{n^2+3}{\sqrt{4+n^7}} \qquad \qquad \sum_{k=3}^\infty \frac{k\cos^2(k)}{1+k^4} \] \[ \sum_{n=1}^\infty \frac{1+\cos(n)}{\mathrm{e}^n} \qquad \qquad \sum_{n=1}^\infty \frac{3}{\sqrt{n}-3} \qquad \qquad \sum_{n=1}^\infty \frac{1}{n^n} \] \[ \sum_{n=1}^\infty \left(1+\frac{1}{n}\right)^2\mathrm{e}^{-n} \qquad \qquad \sum_{k=1}^\infty \frac{\tan\!\left(\frac{1}{k}\right)}{k} \qquad \qquad \sum_{k=1}^\infty \frac{5n+5^n}{3n+3^n} \]