Let $\{a_n\}$ be a positive, convergent sequence. We consider the sequence of partial sum $\{s_n\}: s_n = \sum_{k=1}^n a_n$. Clearly $\{s_n\}$ is strictly increasing and therefore $\sum_{n=1}^\infty s_n$ diverges to $\infty$.
Here's the problem: we want to calculate $\sum_{n=1}^\infty na_n$. Since $a_n = s_n - s_{n-1}$, we have:
$\sum_{n=1}^\infty na_n = \sum_{n=1}^\infty n(s_n-s_{n-1}) = \sum_{n=1}^\infty ns_n - \sum_{n=1}^\infty ns_{n-1} = \sum_{n=1}^\infty ns_n - \sum_{n=1}^\infty (n+1)s_n = -\sum_{n=1}^\infty s_n$
Thus $\sum_{n=1}^\infty s_n = -\sum_{n=1}^\infty na_n$. This doesn't make sense since we can make the series $\sum_{n=1}^\infty na_n$ converge, while $\sum_{n=1}^\infty s_n$ should still diverge. Also, both $\sum_{n=1}^\infty na_n$ and $\sum_{n=1}^\infty s_n$ should be $\geq 0$.
Anything wrong with the above argument?
The problem is with the part where you write $\sum_{n=1}^\infty n(s_n-s_{n-1}) = \sum_{n=1}^\infty ns_n - \sum_{n=1}^\infty ns_{n-1}$. Both "sums" on the right diverge.