Question

Find the radius of convergence for each of the following power series:

1. \[\sum_{k=2}^{\infty} x ^{k} / klog ^{2}(k)\]
2. \[\sum_{n=1}^{\infty} 3^{n}x ^{n} / n!
3. \[\sum_{k=2}^{\infty} x ^{k} / klog(k)\]
4. \[\sum_{k=1}^{\infty} \log(k)x ^{k}\]
5. \[\sum_{n=1}^{\infty} (-1)^{n}x ^{2n+1} / n!((n+1)!2^{2n+1}\]
6. \[1 + x ^{3}/2.3 + x ^{6}/2.3.5.6 + x ^{9}/2.3.5.6.8.9 + ....\]
7. \[\sum_{k=1}^{\infty} x ^{k ^{2}}\]
8. \[\sum_{k=1}^{\infty} x ^{k!}\]