radius of convergence of the power series

Hi, can anyone please help me with this question?

Let $\displaystyle \displaystyle \sum_{n=0}^{\infty} a_n z^n$ be a power series with complex coefficients $\displaystyle a_n$ such that $\displaystyle \displaystyle \sum_{n=0}^{\infty} \mid a_n \mid $converges but $\displaystyle \displaystyle \sum_{n=o}^{\infty} n \mid a_n \mid$ diverges. Prove that the radius of convergence of the power series is 1.

I had some hint that I need to try to prove it by contradiction, first prove $\displaystyle R \geq 1$, then prove R can't be great than 1 by contradiction. But I'm not sure how to do it. Can anyone please help me? Thanks a lot.