# Convergent series and of positive integers and partial sums.

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!!

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Hint: a great part of the problem is done once we justify the computation $$\sum_{n=1}^\infty \sum_{k\geqslant n}a_k=\sum_{k=1}^{\infty}\sum_{n=1}^{k}a_k=\sum_{k=1}^{+\infty}ka_k.$$
That's more than a great part of the problem. That's it. Note your $n$ runs from $1$ to $k$, so you get $k$ and not $k-1$. Pardon me if I'm wrong. –  1015 Mar 25 '13 at 21:47