For (a), you could take the binomial series for and . They both have radius of convergence 1. But their product is the series 1+0+0+0+..., which obviously has infinite radius of convergence.
That won't be enough for (c), where you want the radius of convergence of the product to stay finite. But you could possibly adapt the solution to (a), for example by replacing by . The power series for that function will be the Cauchy product of the series for and , and will still have radius of convergence 1. But when you multiply it by the product is , whose power series has radius of convergence 2.