Suppose and have radii of convergence and respectively. Show that the Cauchy Product converges for .
So . Assume . Put where . Then for large enough. Thus . Thus converges.
Is this correct?
Yes, it is. Notice, however, that this does NOT say that the radius of convergence is min(R1,R2). It is possible that the Cauchy product of two power series converges outside the radius of convergence of one of them.