Do $\Sigma_{n=1}^{\infty} a_n$ and $\Sigma_{n=1}^{\infty} n(a_n -a_{n+1})$ converge simultaneously?
$$\Sigma_{n=1}^{\infty} n(a_n -a_{n+1}) = a_1 - a_2 + 2a_2 -2a_3 + 3a_3-3a_4 + ... = \Sigma_{n=1}^{\infty} a_n$$
It looks like they could possible differ by some term $-a_{n+1}$, but I am not sure how to prove this.
If you write the partial sums up to $n=N$ for the two series you will see that they differ by $NA_{N+1}$. If you assume that $na_n \to 0$ then the two series both converge or both diverge. But in general this is not true as the example by eminem shows. Your mistake is in manipulating infinite sums without even knowing their convergence.