What is the radius of convergence of following series?

Question

suppose that $\sum b_n$ is conditionally convergent but not absolutely convergent.

What is the radius of convergence of the following power series $p(x)=\sum b_nx^n$?

Answer 1

We have $|b_n|\to 0$ (Baby Rudin 3.23). Thus, for $|x|<1$, $\sum |b_n||x|^n$ is convergent since $\sum |x|^n$ is (comparison test). The radius of convergence is therefore $\ge 1$. If the radius $R$ were $> 1$, then we should have absolute convergence for $|x|<R$ (follows from proof of Baby Rudin 3.39). This contradicts the non-absolute convergence for $x=1$. Thus, the radius must be $R=1$.