Let $\sum a_n$ be a convergent series of positive real numbers with sum $s$ and partial sums $s_n=a_1+a_2+\cdots+a_n$. Prove that $\sum na_n$ is convergent if and only if $\sum (s-s_n)$ is convergent.
I have been trying to work this out for a while. I have concluded that since all $a_n$ are positive real numbers and $s_n$ is a partial sum of $s$, that $\sum (s-s_n)$ is always convergent. However, I do not know how to link this to $\sum na_n$. All help is much appreciated!!