Find a sequence $\{a_n\}$ such that both $\sum {a_n}$ and $\sum{\frac{1}{n^2 a_n}}$ converge.

Question

Find a sequence $\{a_n\}$ such that both $\sum_1^\infty {a_n}$ and
$\sum_1^\infty {\frac{1}{n^2 a_n}}$ converge. If no such sequence
exists, prove that.

Actually the question was for $\sum_1^\infty {n{a_n}}$ and $\sum_1^\infty {\frac{1}{n^2 a_n}}$. For this problem; If we suppose that both are convergent, then their multiply $\sum_1^\infty 1/n$ must be convergent and this is a contradiction.

But for $\sum_1^\infty {a_n}$ and $\sum_1^\infty {\frac{1}{n^2 a_n}}$, if we multiply we get $\sum_1^\infty 1/n^2$ which is convergent. I tried to find a sequence such that both the series converge but I couldn't find any but also I couldn't prove that there is no such sequence exists.

Edit. $a_n >0$

Answer 1

Perhaps a bit too trivial?

$a_n=(-1)^n(1/n)$