Maybe this works. Let $a_1,a_2,...$ be the positive terms (there have to be infinitely many of them). Let $b_1,b_2,...$ be the negative terms (there have to be infinitely many of them). Also, $a_1+a_2+.... = \infty$ and $b_1+b_2+... = -\infty$ (because otherwise we have absolute convergence). Pick enough terms so that $a_1+...+a_{k_1}> 1$. Then pick $b_1$. Then pick enough terms so that $a_1+...+a_{k_1}+b_1+a_{k_1+1}+...+a_{k_2} >2$. Then pick $b_2$. And keep on going.