Conditions for Alternating series to diverge

Question

Series $(-1)^nb_n$ converges conditionally if the series $\{b_n\}_{n=1}^{\infty}$ diverges but two conditions are satisfied:

1. the series is non increasing .
2. $\displaystyle\lim_{n\to \infty}{b_n} = 0$

I want to know if the 1st condition(it is not non increasing ) is not satisfied , does it mean the series diverges ?

Answer 1

Not necessarily. If $b_n$ is defined as $b_n=\begin{cases}\frac{1}{n} &\text{if $n=4k$ or $n=4k+1$ for some $k\in\mathbb{N}$}\\ \frac{1}{2^n} &\text{if $n=4k+2$ or $n=4k+3$ for some $k\in\mathbb{N}$}\end{cases}$ Then the alternating series $\sum (-1)^n b_n$ converges but $b_n$ is not decreasing.