An absolutely summable series but not 1-summable?

Question

I am trying to find an example of an absolutely summable series ($\Sigma|{a_j}|<\infty$) but not 1-summable series ($\Sigma j|{a_j}|<\infty$).

I know that all 1-summable series are absolutely-summable. I proved it.
But what about the converse? Is the converse also true, or is there a counter example?

Answer 1

$a_j =\frac 1 {j^{2}}$ has this property.