Since b_n≥0 for all n, the sequence (t_n) of partial sums of ∑ b_n is an increasing sequence. In part (a) this sequence is bounded above by the sum of series ∑ a_n , so (t_n) converges by the monotone convergence theorem (16.3). Thus ∑ b_n converges. In part (b) the sequence (t_n) must be unbounded, for otherwise ∑ a_n would have to converge. But then lim t_n=+∞ by Theorem 16.8, so that ∑ b_n=+∞.