Does Using Maximum Coefficients Determine the Smallest Radius of Convergence?

[aaaa202]

Homework Statement

Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B)

Homework Equations

My idea is to use that the series Ʃ(lanl+lbnl)x^n has convergence radius min(A,B) and use that lanl+lbnl≥dn. Do you agree that this is a good idea from a rigorous perspective? Last assigment I really got punished for not being rigorous enough, so I want to make sure this time, that I do it properly.

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B)

Homework Equations

My idea is to use that the series Ʃ(lanl+lbnl)x^n has convergence radius min(A,B) and use that lanl+lbnl≥dn. Do you agree that this is a good idea from a rigorous perspective? Last assigment I really got punished for not being rigorous enough, so I want to make sure this time, that I do it properly.

That will show the radius of convergence of $\sum d_n$ is at least min(A,B). You would still have to show it isn't greater than that.