Example of Cauchy product of two series with radius of convergence $\rho'> \min\{\rho_1,\rho_2\}$

Question

Can anyone suggest a (relatively simple) example of two real power series $\sum_{n \geq 0} a_n x^n$ and $\sum_{n \geq 0} b_n x^n$, with radii of convergence rispectively $\rho_1$ and $\rho_2$, whose Cauchy product series has a radius of convergence $\rho'> \min\{\rho_1,\rho_2\}$?

Answer 1

Here's a simple but somewhat contrived example:

Consider the power series $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$ with $a_n=1$ for all $n$, and the polynomial $1-x$ with $b_0=1,b_1=-1$ and $b_n=0$ for $n\geq 2$. Note that $\rho_1=1$ and $\rho_2=\infty$.

Then the coefficients of the Cauchy product are $c_0=1$ and $$ c_n=\sum_{k=0}^na_kb_{n-k}=a_{n-1}b_1+a_{n}b_0=0 $$ for $n\geq 1$. Thus $\rho^{\prime}=\infty>\min\{\rho_1,\rho_2\}$.