A question on asymptotics of a sequence and that of corresponding series

Question

$\beta_n$ is a sequence of positive real numbers.It is known that $\beta_n = O(1/n)$ and $\beta_n\ne o(1/n)$. What can we say about the series $$\sum_{k=0}^n \beta_k$$

Answer 1

The series may converge or diverge. Consider the examples

$$ \beta_n = \frac{1}{n} $$

and

$$ \beta_n = \begin{cases} 2^{-n} & \text{if } n \neq 2^m \text{ for any } m \in \mathbb Z, \\ 2^{-m} & \text{if } n = 2^m \text{ for some } m \in \mathbb Z. \end{cases} $$