Series convergence $sumlimits_{k = 1}^{+infty} k a_k^2$

I found a nice problem about series convergence. And would like to share it.

Suppose series $\sum\limits_{k = 1}^{+\infty} a_k$ converges absolutely.

1) Found an example when $\sum\limits_{k = 1}^{+\infty} k a_k^2$ diverges.

2) Suppose that $a_k$ is a non increasing sequence. Is it true that $\sum\limits_{k = 1}^{+\infty} k a_k^2$ converges?

3) Is it true that if $a_k$ is a non increasing then $\sum\limits_{k = 1}^{+\infty}k^2 a_k^2$ converges?