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.

  • 2
    $\begingroup$ Use the root test, noting that limsup of a product $\le$ product of the limsups. $\endgroup$ – Robert Israel May 9 at 15:40
  • $\begingroup$ Why can’t you use lim sup? How do you define the radius of convergence then? $\endgroup$ – rhombicosicodecahedron May 9 at 15:51
  • $\begingroup$ 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. $\endgroup$ – Bradley Mangham May 9 at 15:56
  • $\begingroup$ If none of $a_n,b_n$ for any n are zero, then you can use the ratio test. If that is allowed. $\endgroup$ – rhombicosicodecahedron May 9 at 16:13
  • 1
    $\begingroup$ @rhombicosicodecahedron The most natural way to define the ROC is not through the root test, but as $\sup\{r\ge 0: \sum_n |a_n|r^n<\infty\}.$ $\endgroup$ – zhw. May 9 at 19:55

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