Let $\sum_{n = 0}^{\infty} a_nz^n$ and $\sum_{n = 0}^{\infty} b_nz^n$ be two power series, each with radius of convergence 1. How large can the radius of convergence of their Cauchy product, $\sum_{n = 0}^{\infty} c_nz^n$ be? (to be clear, $c_n = \sum_{k = 0}^{n} a_kb_{n – k}$).
I know that the radius of convergence of the Cauchy product is lower bounded by the minimum of the radius of convergence of either power series, but this question appears to ask for an upper bound, which leaves me rather puzzled.
Best Answer
The convergence radius of the Cauchy product $\sum c_n x^n$ can be infinite when the series $\sum a_n x^n$ and $\sum b_n x^n$ have finite convergence radii.
For example,
$$1 = \frac{1+x}{1-x} \cdot \frac{1-x}{1+x} = \left((1 + x) \sum_{n=0}^\infty x^n \right) \left((1 - x) \sum_{n=0}^\infty (-x)^n \right) \\= \left(1 + 2\sum_{n=0}^\infty x^n \right) \left(1 +2 \sum_{n=0}^\infty (-x)^n \right)$$
The series on the RHS each have convergence radius $1$ but the Cauchy product
$$\sum_{n=0}^\infty c_n x^n = 1 + \sum_{n=1}^\infty 0 \cdot x^n$$
has an infinite radius of convergence.