Power series product convergence

[Hill]

Homework Statement

Give an example of a pair of origin-centered power series, say P(z) and Q(z), such that the disc of convergence for the product P(z)Q(z) is larger than either of the two discs of convergence for P(z) and Q(z).

Relevant Equations

$\frac 1 {1-z} = 1+z+z^2+z^3+\cdots$

I take $$P(z)=\frac {1-z}{5-z} = \frac 1 5 -\frac 4 {25} z - \frac 4 {125} z^2 - \cdots$$ which has radius of convergence 5, and $$Q(z)=\frac {5-z} {1-z} = 5+4z+4z^2+\cdots$$ which has radius of convergence 1.
$P(z)Q(z)=1$ converges everywhere.
Is this correct? If so, do you think it's a good example or rather a dirty trick, or both?

 

[mfb]

I think that is a good example.

 

[FactChecker]

I like it. It's much simpler and more convincing than any examples that I started to think of.