Dear all, Let the positive series $\sum_{n=1}^\infty a_n$ be divergent, and $\lim_{n\to\infty}\frac{a_n}{A_n}=0,$ where $A_n=\sum_{k=1}^n a_k$. Prove that the radius of convergence of the power series $\sum_{n=1}^\infty a_nx^n$ equals $1$.
My thought: Since $\sum_{n=1}^\infty a_n$ diverges, we have the radius of convergence $R\leq 1.$ However, I could not see how to show $R<1$ is impossible. 3x!