If $\displaystyle a_k \to 0$ as $\displaystyle k \to \infty$, and $\displaystyle \displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $\displaystyle a_k \downarrow 0$ as $\displaystyle k \to \infty$.

I think this is false and my example is $\displaystyle (-1)^k \frac{1}{k^2}$. What you think?