What could be an example where multiplying two power series gives a bigger radius that is not infinity

Question

I'm trying to think of an example of $\sum_{n=0}^{\infty}a_nx^n, \sum_{n=0}^{\infty}b_nx^n$ with radius of convergence $R_1,R_2$ such that when we multiply them we get $\sum_{n=0}^{\infty}c_nx^n, c_n = \sum_{i+j=n}a_ib_j=\sum_{k=0}^{n} a_jb_{n-j}$ with radius $R$ such that $R>R_1$, $R>R_2$ and $R<\infty$. What could be an easy example for that?

Answer 1

Consider the Taylor series $\sum_{k\geq0} a_k\,x^k$ of the functions $$f(x):={2-x\over(1-x)(3-x)},\qquad g(x):={1-x\over 2-x}\ .$$ Then $R_f=1$, $\>R_g=2$, and $R_{fg}=3$, because the radius of convergence for the Taylor series at $0$ is equal to the distance from $0$ to the nearest singularity.