Exercise 16. Determine the radius of convergence of the series $\sum_{n=1}^{\infty} a_n z^n$ when: (1) $a_n=(\log n)^2$ (2) $a_n=n$ ! (3) $a_n=\frac{n^2}{4^n+3 n}$ (4) $a_n=(n !)^3 /(3 n)$ ! (5) Find the radius of convergence of the hypergeometric series \[ F(\alpha, \beta, \gamma ; z)=1+\sum_{n=1}^{\infty} \frac{\alpha(\alpha+1) \cdots(\alpha+n-1) \beta(\beta+1) \cdots(\beta+n-1)}{n ! \gamma(\gamma+1) \cdots(\gamma+n-1)} z^n . \] Here $\alpha, \beta \in \mathbb{C}$ and $\gamma \neq 0,-1,-2, \cdots$ (6) Find the radius of convergence of the Bessel function of order $r$ : \[ J_r=\left(\frac{z}{2}\right)^r \sum_{n=0}^{\infty} \frac{(-1)^n}{n !(n+r) !}\left(\frac{z}{2}\right)^{2 n}, \] where $r$ is a positive integer.