$\{b_n\}$ is increasing and $\lim_{n \to \infty} b_n = 0$

$\{b_n\}$ is decreasing and $\lim_{n \to \infty} b_n = 0$

$\{b_n\}$ is bounded and $\lim_{n \to \infty} b_n = 1$

$\{b_n\}$ is positive and $\sum b_n$ diverges