An example of a power series that has a radius of convergence of 3 The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if this is a little simplistic but I really can't think of anything and am having a hard time with this.

Taking the center of the desired power series at origin, the domain of convergence must be $|z|<3$. Indeed, you need a power series of the form $f(z)=\sum_{n=0}^{\infty}$$a_n.z^n$ which converges uniformly if $|z/3|<1$. One such choice is $f(z)=\frac{1}{3-z}=\sum_{n=0}^{\infty}$$\frac{1}{3^{n+1}}.z^n$.
