sigma ( An ) converges and An >0
Show that there exist Bn with Bn/An tends to 0 and Sigma( Bn) converges
If $\displaystyle \sum_{n=0}^{\infty}a_n$ converges then it absolutely converges, which means $\displaystyle \sum_{n=0}^{\infty}a_n^2$ will converge. Take $\displaystyle b_n = a_n^2$.