# Radius of Convergence of the 'product' of two power series of the same radius of convergence

I have come across a problem involving the radius of convergence of power series that I can't solve for the life of me, any help would be greatly appreciated. I cannot use the lim sup method to solve this.

The problem is as follows.

Prove that if $$\sum_{n=0}^\infty a_nx^n$$ has radius of convergence $$R_1 > 1$$ and $$\sum_{n=0}^\infty b_n x^n$$ has radius of convergence $$R_2 > 1$$ then the radius of convergence R of $$\sum_{n=0}^\infty a_nb_nx^n$$ is at least $$R_1R_2$$

Thank you.

• Use the root test, noting that limsup of a product $\le$ product of the limsups. – Robert Israel May 9 at 15:40
• Why can’t you use lim sup? How do you define the radius of convergence then? – rhombicosicodecahedron May 9 at 15:51
• We have covered it and defined the R.O.C that way but the professor stated not to use it. Ill ask him if it's intended for a different question but as far as I'm aware it is intentional. – Bradley Mangham May 9 at 15:56
• If none of $a_n,b_n$ for any n are zero, then you can use the ratio test. If that is allowed. – rhombicosicodecahedron May 9 at 16:13
• @rhombicosicodecahedron The most natural way to define the ROC is not through the root test, but as $\sup\{r\ge 0: \sum_n |a_n|r^n<\infty\}.$ – zhw. May 9 at 19:55