We can find a stricly increasing sequence of integer (n_k) such that q_{n_k}\geq k. Put a_{n_k}=1/(kq_{n_k}) and a_j=0 if j\neq n_k for all k.
q_1, q_2, ... , q_n, ... is a sequence of real numbers such that the limit of q_n, as n goes towards infinity, is +infinity.
Show that we can find a sequence a_1, a_2, ... , a_n, ... sych that the sum(a_n) is convergent but the sum(a_n*q_n) is divergent.