If R and R' are respectively the radius of convergence of the series
then, the radius of convergence R'' of the product series
satisfies
Now, use that both series are absolutely convergent in | x | < R'' and apply a well known theorem.
A(x) = sum(n=0 -> infinity) of a(subn)x^n
B(x) = sum(n=0 -> infinity) of b(subn)x^n
A(x) * B(x) = sum(n=0 -> infinity) of a(subn)x^n * sum(n=0 -> infinity) of b(subn)x^n
= sum(k=0 -> infinity) of (a(subk)*b(sub(n-k))x^n (by definition of mulitiplication)
= ?
= sum(k=0 -> infinity) of (b(subk)*a(sub(n-k))x^n
= sum(n=0 -> infinity) of b(subn)x^n * sum(n=0 -> infinity) of a(subn)x^n
= B(x) * A(x)
I can't figure out the missing step... Any advice? Thanks!