Dear all, Let the positive series $\displaystyle \sum_{n=1}^\infty a_n$ be divergent, and $\displaystyle \lim_{n\to\infty}\frac{a_n}{A_n}=0,$ where $\displaystyle A_n=\sum_{k=1}^n a_k$. Prove that the radius of convergence of the power series $\displaystyle \sum_{n=1}^\infty a_nx^n$ equals $\displaystyle 1$.